how to write a hypothesis for an interaction

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8.6 - interaction effects, example 8-4: depression treatments section  .

Now that we've clarified what additive effects are, let's take a look at an example where including " interaction terms " is appropriate.

Some researchers (Daniel, 1999) were interested in comparing the effectiveness of three treatments for severe depression. For the sake of simplicity, we denote the three treatments A, B, and C. The researchers collected the following data ( Depression Data ) on a random sample of n = 36 severely depressed individuals:

• \(y_{i} =\) measure of the effectiveness of the treatment for individual i
• \(x_{i1} =\) age (in years) of individual i
• \(x_{i2} = 1\) if individual i received treatment A and 0, if not
• \(x_{i3} = 1\) if individual i received treatment B and 0, if not

A scatter plot of the data with treatment effectiveness on the y -axis and age on the x -axis looks like this:

The blue circles represent the data for individuals receiving treatment A, the red squares represent the data for individuals receiving treatment B, and the green diamonds represent the data for individuals receiving treatment C.

In the previous example, the two estimated regression functions had the same slopes —that is, they were parallel. If you tried to draw three best-fitting lines through the data of this example, do you think the slopes of your lines would be the same? Probably not! In this case, we need to include what are called " interaction terms " in our formulated regression model.

A (second-order) multiple regression model with interaction terms is:

\(y_i=\beta_0+\beta_1x_{i1}+\beta_2x_{i2}+\beta_3x_{i3}+\beta_{12}x_{i1}x_{i2}+\beta_{13}x_{i1}x_{i3}+\epsilon_i\)

and the independent error terms \(\epsilon_i\) follow a normal distribution with mean 0 and equal variance \(\sigma^{2}\). Perhaps not surprisingly, the terms \(x_{i} x_{i2}\) and \(x_{i1} x_{i3}\) are the interaction terms in the model.

Let's investigate our formulated model to discover in what way the predictors have an " interaction effect " on the response. We start by determining the formulated regression function for each of the three treatments. In short —after a little bit of algebra (see below) —we learn that the model defines three different regression functions —one for each of the three treatments:

So, in what way does including the interaction terms, \(x_{i1} x_{i2}\) and \(x_{i1} x_{i3}\), in the model imply that the predictors have an " interaction effect " on the mean response? Note that the slopes of the three regression functions differ —the slope of the first line is \(\beta_1 + \beta_{12}\), the slope of the second line is \(\beta_1 + \beta_{13}\), and the slope of the third line is \(\beta_1\). What does this mean in a practical sense? It means that...

• the effect of the individual's age \(\left( x_1 \right)\) on the treatment's mean effectiveness \(\left(\mu_Y \right)\) depends on the treatment \(\left(x_2 \text{ and } x_3\right)\), and ...
• the effect of treatment \(\left(x_2 \text{ and } x_3\right)\) on the treatment's mean effectiveness \(\left(\mu_Y \right)\) depends on the individual's age \(\left( x_1 \right)\).

In general, then, what does it mean for two predictors " to interact "?

• Two predictors interact if the effect on the response variable of one predictor depends on the value of the other .
• A slope parameter can no longer be interpreted as the change in the mean response for each unit increase in the predictor, while the other predictors are held constant.

And, what are " interaction effects "?

A regression model contains interaction effects if the response function is not additive and cannot be written as a sum of functions of the predictor variables. That is, a regression model contains interaction effects if:

\(\mu_Y \ne f_1(x_1)+f_1(x_1)+ \cdots +f_{p-1}(x_{p-1})\)

For our example concerning treatment for depression, the mean response:

\(\mu_Y=\beta_0+\beta_1x_{1}+\beta_2x_{2}+\beta_3x_{3}+\beta_{12}x_{1}x_{2}+\beta_{13}x_{1}x_{3}\)

can not be separated into distinct functions of each of the individual predictors. That is, there is no way of "breaking apart" \(\beta_{12} x_1 x_2 \text{ and } \beta_{13} x_1 x_3\) into distinct pieces. Therefore, we say that \(x_1 \text{ and } x_2\) interact, and \(x_1 \text{ and } x_3\) interact.

In returning to our example, let's recall that the appropriate steps in any regression analysis are:

• Model formulation
• Model estimation
• Model evaluation

So far, within the model-building step, all we've done is formulate the regression model as:

We can use Minitab —or any other statistical software for that matter —to estimate the model. Doing so, Minitab reports:

Regression Equation

y = 6.21 + 1.0334 age + 41.30 x2+ 22.71 x3 - 0.703 agex2 - 0.510 agex3

Now, if we plug the possible values for \(x_2 \text{ and } x_3\) into the estimated regression function, we obtain the three "best fitting" lines —one for each treatment (A, B, and C) —through the data. Here's the algebra for determining the estimated regression function for patients receiving treatment A.

Doing similar algebra for patients receiving treatments B and C, we obtain:

And, plotting the three "best fitting" lines, we obtain:

What do the estimated slopes tell us?

• For patients in this study receiving treatment A, the effectiveness of the treatment is predicted to increase by 0.33 units for every additional year in age.
• For patients in this study receiving treatment B, the effectiveness of the treatment is predicted to increase by 0.52 units for every additional year in age.
• For patients in this study receiving treatment C, the effectiveness of the treatment is predicted to increase by 1.03 units for every additional year in age.

In short, the effect of age on the predicted treatment effectiveness depends on the treatment given. That is, age appears to interact with treatment in its impact on treatment effectiveness. The interaction is exhibited graphically by the "nonparallelness" (is that a word?) of the lines.

Of course, our primary goal is not to draw conclusions about this particular sample of depressed individuals, but rather about the entire population of depressed individuals. That is, we want to use our estimated model to draw conclusions about the larger population of depressed individuals. Before we do so, however, we first should evaluate the model.

The residuals versus fits plot:

exhibits all of the "good" behavior, suggesting that the model fits well, there are no obvious outliers, and the error variances are indeed constant. And, the normal probability plot:

exhibits a linear trend and a large P -value, suggesting that the error terms are indeed normally distributed.

Having successfully built —formulated, estimated, and evaluated —a model, we now can use the model to answer our research questions. Let's consider two different questions that we might want to be answered.

First research question. For every age, is there a difference in the mean effectiveness for the three treatments? As is usually the case, our formulated regression model helps determine how to answer the research question. Our formulated regression model suggests that answering the question involves testing whether the population regression functions are identical.

That is, we need to test the null hypothesis \(H_0 \colon \beta_2 = \beta_3 =\beta_{12} = \beta_{13} = 0\) against the alternative \(H_A \colon\) at least one of these slope parameters is not 0.

We know how to do that! The relevant software output:

Analysis of Variance

tells us that the appropriate partial F -statistic for testing the above hypothesis is:

\(F=\frac{(803.8+1.19+375+328.42)/4}{15.4}=24.49\)

And, Minitab tells us:

F Distribution with 4 DF in Numerator and 30 DF in denominator

that the probability of observing an F -statistic —with 4 numerator and 30 denominator degrees of freedom —less than our observed test statistic 24.49 is > 0.999. Therefore, our P -value is < 0.001. We can reject our null hypothesis. There is sufficient evidence at the \(\alpha = 0.05\) level to conclude that there is a significant difference in the mean effectiveness for the three treatments.

Second research question. Does the effect of age on the treatment's effectiveness depend on the treatment? Our formulated regression model suggests that answering the question involves testing whether the two interaction parameters \(\beta_{12} \text{ and } \beta_{13}\) are significant. That is, we need to test the null hypothesis \(H_0 \colon \beta_{12} = \beta_{13} = 0\) against the alternative \(H_A \colon\) at least one of the interaction parameters is not 0.

The relevant software output:

\(F=\dfrac{(375+328.42)/2}{15.4}=22.84\)

F Distribution with 2 DF in Numerator and 30 DF in denominator

that the probability of observing an F -statistic — with 2 numerator and 30 denominator degrees of freedom — less than our observed test statistic 22.84 is > 0.999. Therefore, our P -value is < 0.001. We can reject our null hypothesis. There is sufficient evidence at the \(\alpha = 0.05\) level to conclude that the effect of age on the treatment's effectiveness depends on the treatment.

A model with an interaction term Section  

• The formulated regression function for patients receiving treatment B.
• The formulated regression function for patients receiving treatment C.

\(\mu_Y = \beta_0+\beta_1x1+\beta_2 x_2+\beta_3 x_3+\beta_{12}x_1 x_2+\beta_{13} x_1 x_3\)

\(\mu_Y = \beta_0+\beta_1 x_1+\beta_2(0)+\beta_3(1)+\beta_{12} x_1(0)+\beta_{13} x_1(1) = (\beta_0+\beta_3)+(\beta_1+\beta_{13})x_1\)

\(\mu_Y = \beta_0+\beta_1 x_1+\beta_2(0)+\beta_3(0)+\beta_{12} x_1(0)+\beta_{13} x_1(0) = \beta_0+\beta_1 x_1\)

Treatment B, \(x_2 = 0 , x_3 = 1\), so

Treatment C, \(x_2 = 0 , x_3 = 0\), so

For the depression study, plug the appropriate values for \(x_2 \text{ and } x_3\) into the estimated regression function and perform the necessary algebra to determine:

• The estimated regression function for patients receiving treatment B.
• The estimated regression function for patients receiving treatment C.

\(\hat{y} = 6.21 + 1.0334 x_1 + 41.30 x_2+22.71 x_3 - 0.703 x_1 x_2 - 0.510 x_1 x_3\)

\(\hat{y} = 6.21 + 1.0334 x_1 + 41.30(0) + 22.71(1) - 0.703 x_1(0) - 0.510 x_1(1) = (6.21 + 22.71)+(1.0334 - 0.510) x_1 = 28.92 + 0.523 x_1\)

\(\hat{y} = 6.21 + 1.0334 x_1 + 41.30(0) + 22.71(0) - 0.703 x_1(0) - 0.510 x_1(0) = 6.21 + 1.033 x_1\)

For the first research question that we addressed for the depression study, show that there is no difference in the mean effectiveness between treatments B and C, for all ages, provided that \(\beta_3 = 0 \text{ and } \beta_{13} = 0\). ( HINT : Follow the argument presented in the chalk-talk comparing treatments A and C.)

\(\mu_Y|\text{Treatment B} - \mu_Y|\text{Treatment C} = (\beta_0 + \beta_3)+(\beta_1 + \beta_{13}) x_1 - (\beta_0 + \beta_1 x_1) = \beta_3 + \beta_{13} x_1 = 0\), if \(\beta_3 = \beta_{13} = 0\)

A study of atmospheric pollution on the slopes of the Blue Ridge Mountains (Tennessee) was conducted. The Lead Moss data contains the levels of lead found in 70 fern moss specimens (in micrograms of lead per gram of moss tissue) collected from the mountain slopes, as well as the elevation of the moss specimen (in feet) and the direction (1 if east, 0 if west) of the slope face.

• Write the equation of a second-order model relating mean lead level, E ( y ), to elevation \(\left(x_1 \right)\) and the slope face \(\left(x_2 \right)\) that includes an interaction between elevation and slope face in the model.
• Graph the relationship between mean lead level and elevation for the different slope faces that are hypothesized by the model in part a.
• In terms of the β's of the model in part a, give the change in lead level for every one-foot increase in elevation for moss specimens on the east slope.
• Fit the model in part a to the data using an available statistical software package. Is the overall model statistically useful for predicting lead level? Test using \(α = 0.10\).
• Write the estimated equation of the model in part a relating mean lead level, E ( y ), to elevation \(\left(x_1 \right)\) and slope face \(\left(x_2 \right)\).

Since the p-value for testing whether the overall model is statistically useful for predicting lead level is 0.857, we conclude that this model is not statistically useful.

(e) The estimated equation is shown in the Minitab output above, but since the model is not statistically useful, this equation doesn’t do us much good.

\(\mu_Y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_{12} x_1 x_2\)

East slope, \(x_2=1\), \(\mu_Y = \beta_0 + \beta_1 x_1 + \beta_2(1) + \beta_{12} x_1(1) = (\beta_0 + \beta_2)+(\beta_1 + \beta_{12}) x_1\), so average lead level changes by \(\beta_1 + \beta_{12}\) micrograms of lead per gram of moss tissue for every one foot increase in elevation for moss specimens on the east slope.

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• How to Write a Strong Hypothesis | Guide & Examples

How to Write a Strong Hypothesis | Guide & Examples

Published on 6 May 2022 by Shona McCombes .

A hypothesis is a statement that can be tested by scientific research. If you want to test a relationship between two or more variables, you need to write hypotheses before you start your experiment or data collection.

Table of contents

What is a hypothesis, developing a hypothesis (with example), hypothesis examples, frequently asked questions about writing hypotheses.

A hypothesis states your predictions about what your research will find. It is a tentative answer to your research question that has not yet been tested. For some research projects, you might have to write several hypotheses that address different aspects of your research question.

A hypothesis is not just a guess – it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations, and statistical analysis of data).

Variables in hypotheses

Hypotheses propose a relationship between two or more variables . An independent variable is something the researcher changes or controls. A dependent variable is something the researcher observes and measures.

In this example, the independent variable is exposure to the sun – the assumed cause . The dependent variable is the level of happiness – the assumed effect .

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Step 1: ask a question.

Writing a hypothesis begins with a research question that you want to answer. The question should be focused, specific, and researchable within the constraints of your project.

Step 2: Do some preliminary research

Your initial answer to the question should be based on what is already known about the topic. Look for theories and previous studies to help you form educated assumptions about what your research will find.

At this stage, you might construct a conceptual framework to identify which variables you will study and what you think the relationships are between them. Sometimes, you’ll have to operationalise more complex constructs.

Step 3: Formulate your hypothesis

Now you should have some idea of what you expect to find. Write your initial answer to the question in a clear, concise sentence.

Step 4: Refine your hypothesis

You need to make sure your hypothesis is specific and testable. There are various ways of phrasing a hypothesis, but all the terms you use should have clear definitions, and the hypothesis should contain:

• The relevant variables
• The specific group being studied
• The predicted outcome of the experiment or analysis

Step 5: Phrase your hypothesis in three ways

To identify the variables, you can write a simple prediction in if … then form. The first part of the sentence states the independent variable and the second part states the dependent variable.

In academic research, hypotheses are more commonly phrased in terms of correlations or effects, where you directly state the predicted relationship between variables.

If you are comparing two groups, the hypothesis can state what difference you expect to find between them.

Step 6. Write a null hypothesis

If your research involves statistical hypothesis testing , you will also have to write a null hypothesis. The null hypothesis is the default position that there is no association between the variables. The null hypothesis is written as H 0 , while the alternative hypothesis is H 1 or H a .

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

A hypothesis is not just a guess. It should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations, and statistical analysis of data).

A research hypothesis is your proposed answer to your research question. The research hypothesis usually includes an explanation (‘ x affects y because …’).

A statistical hypothesis, on the other hand, is a mathematical statement about a population parameter. Statistical hypotheses always come in pairs: the null and alternative hypotheses. In a well-designed study , the statistical hypotheses correspond logically to the research hypothesis.

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From the Editors: Explaining interaction effects within and across levels of analysis

• Published: 05 November 2014
• Volume 45 , pages 1063–1071, ( 2014 )

Cite this article

• Ulf Andersson 1 ,
• Alvaro Cuervo-Cazurra 2 &
• Bo Bernhard Nielsen 3  

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Many manuscripts submitted to the Journal of International Business Studies propose an interaction effect in their models in an effort to explain the complexity and contingency of relationships across borders. In this article, we provide guidance on how best to explain the interaction effects theoretically within and across levels of analysis. First, in the case of interactions within the same level of analysis, we suggest that authors provide an explanation of the mechanisms that link the main independent variable to the dependent variable, and then explain how the interaction variable modifies these mechanisms. Moreover, to ensure that the arguments are theoretically complete, we suggest that authors theoretically rule out the potential reverse interaction effect between the main variable and moderating variable. Second, in the case of interactions across levels of analysis, we suggest that authors identify the cross-level nature of the moderating relationships, specify the level of analysis of the main relationship and the nested nature of the cross-level influences, and theoretically explain these cross-level influences. Additionally, we suggest that authors pay particular attention to nesting in order to theoretically rule out reverse interactions.

Avoid common mistakes on your manuscript.

INTRODUCTION

As editors, we are increasingly seeing papers with interaction effects – also known as multiplicative effects, product terms or moderation effects – that benefit from the powerful statistical analyses now available to scholars. Such research strategy has the potential to yield new theoretical insights that may advance the international business (IB) field. However, incorporating interaction effects is challenging because it identifies new and complex relationships that needs to be adequately explained. To help authors, in this editorial we provide suggestions for how best to explain the theoretical mechanisms Footnote 1 behind proposed interaction effects in order to clarify the theoretical contribution of their studies. We go beyond statistical explanations of interaction effects and their detection (see, e.g., Aguinis & Gottfredson, 2010 ; Jaccard & Turrisi, 2003 ; Shieh, 2009 ), which, depending on how the variables are measured and the type of statistical method used, can be quite challenging.

We discuss two types of interaction effects: within and across levels of analysis. First, for interactions within levels of analysis, we suggest that authors first provide an explanation of the theoretical mechanisms that link the main independent variable to the dependent variable, and then explain how and why the interaction variable modifies these theoretical mechanisms. Additionally, we suggest that authors theoretically rule out the existence of a reverse interaction effect in which the independent variable is actually affecting the relationship between the moderator and dependent variable. Second, for interactions across levels of analysis, we suggest that authors first identify the level of analysis of the main relationship, then specify the cross-level nature of the moderating relationships, before clarifying the hierarchy and nature of theoretical nesting. In addition, we propose that authors theoretically explain the multilevel influences, separating the justification of the cross-level interaction effect from the explanation of the cross-level direct influences.

EXPLAINING INTERACTION EFFECTS

Interaction effects.

Generally, interaction is said to occur when the effect of an independent variable ( X ) on a dependent variable ( Y ) varies across levels of a moderating variable ( Z ). Identifying and specifying relevant and important interaction effects pertaining to relations between independent and dependent variables is at the heart of theory in social science ( Cohen, Cohen, West, & Aiken, 2003 ) and indicates the maturity and sophistication of a field of inquiry ( Aguinis, Boik, & Pierce, 2001 ). Interactions provide researchers with the ability to enrich our understanding of economic and social relationships by establishing the conditions under which such relationships apply, or are stronger or weaker. As such, interactions enable the extension of well-known relationships to contexts that the original research did not consider, and they also help provide more detailed predictions about the relationships, going beyond the simplistic argument “it depends”. However, merely detecting a statistically significant effect of the interaction between independent and moderating variables on the dependent variable is not sufficient to be considered a contribution to the literature. The interaction effect has to be explained, and there must be theoretical arguments for why including this interaction results in better theory.

Research questions involving moderators typically address “when” or “under what conditions” an independent variable most strongly influences an outcome variable. More specifically, a moderator is a variable that alters the nature or strength of the relationship between an independent and an outcome variable ( Baron & Kenny, 1986 ). The distinction between circumstances where the nature of the relationship of X on Y varies as a function of Z (differential prediction) vs the strength of the relationship of X on Y varies as a function of Z (differential validity) is important for several reasons. First, only differential prediction is appropriately tested with moderated multiple regression, which is the statistical test typically employed in moderation studies ( Carte & Russell, 2003 ). Differential validity is typically tested via subgroup moderation: the sample is split into two or more groups based on the level of the moderator variable, and t -tests of the correlation coefficients and χ 2 tests are performed to assess the strength of the moderation effect and differences among groups. Second, the language and argumentation employed in moderation hypotheses is often inaccurate in relation to the actual tests performed. For instance, if a researcher asserts that “the strength of the multinationality–performance relationship depends on the level of product diversification”, then he/she must report differences in strength of the multinationality–performance relationship (i.e., r multinationality–performance ) across levels of product diversification rather than the often-reported differences in the slope (nature) of the multinationality–performance relationship across levels of product diversification. Scholars must specify the role of the moderation a priori and make sure that the language, theoretical argumentation, and ensuing empirical tests match.

The choice of the moderating variable should be based on a specific theory regarding why, or under what conditions, a given relationship may be significantly influenced for some types of firms, teams, or individuals rather than for others. This choice is important because it drives the specific type of interaction that needs to be explained. First, there are interactions between two continuous variables, which can take three typical patterns ( Cohen et al., 2003 : 285–286): (a) enhancing interactions, in which both the predictor and moderator affect the outcome variable in the same direction and together they have a stronger effect than a merely additive one; (b) buffering interactions, in which the moderator variable weakens the effect of the predictor variable on the outcome; and (c) antagonistic interactions, in which the predictor and moderator have the same effect on the outcome but the interaction is in the opposite direction. Second, there are interactions between a categorical variable and a continuous variable, which can take two different patterns: (a) existence interaction, when an independent variable is positively related to the dependent variable for one particular group but unrelated for another group; and (b) competing interactions, when an independent variable is positively related to the dependent variable for one particular group but it is negatively related for another.

The distinction between the different patterns of interaction has important implications for theory, as the selection of the particular type of interaction should be driven by the specific nature of the concepts analyzed rather than by the particular measurement of the variables used in the statistical analysis. Although all interaction types have the potential for advancing theory, the buffering and antagonistic interactions between continuous variables, and the competing interaction between a categorical variable and continuous variable, hold the greatest potential because they are more likely to challenge existing theory.

Challenges in Explaining Interactions

Regardless of the particular type of interaction proposed, the following are some of the common challenges we find in the explanation of interaction effects in many initial drafts of manuscripts that propose an interaction.

First, there is often no explanation of, or indeed theoretical justification for, the direct effect. Far too often, manuscripts simply start with an explanation of the interaction effect. One reason for this may be that the authors think that the novelty of the paper resides in the interaction effect, because the direct effect has been explained in detail and tested before. However, this approach is problematic because the theoretical mechanism explaining the baseline argument remains unspecified. As a result, it becomes unclear what baseline effect the interaction is supposed to modify. This is particularly problematic because many management and IB phenomena can be explained from many alternative theoretical perspectives; the different mechanisms that link the independent variable to the dependent variable may be rooted in different theories, which offer different logics even if they end up resulting in the same hypothesized relationship. For instance, the relationship between multinationality and performance can be explained from the theoretical standpoints of internalization theory ( Buckley & Casson, 1976 ), behavioral theory of internationalization ( Johanson & Vahlne, 1977 ), or resource-based view ( Penrose, 1959 ), among others. While related, each theory argues for different mechanisms explaining the performance consequences of multinationality, so the explanation of the moderating variable’s influence will also differ depending on the theory considered. For example, if one considers product diversification as a moderator, the challenge is to decide which mechanism it changes: product diversification may alter the costs of multinationality, the risks of multinationality, or the benefits from leveraging firm resources across markets; the choice depends on the theory used to explain the baseline multinationality–performance relationship. Moreover, different theories may actually specify opposing main effects in certain cases; in such cases, failure to specify the nature and direction of the main relationship renders any theorizing or interpretation of the interaction effects ambiguous at best. Such instances may be particularly problematic because even when the direct effect is not statistically significant while the interaction is, the direct relationships must be theoretically justified and described in order for the interaction to make sense.

Second, many manuscripts explain the direct effect of the moderating variable on the dependent variable rather than the impact of the moderating variable on the relationship between the independent and dependent variables. Since manuscripts proposing interaction effects often introduce a new variable, authors often start by defining the new variable and then provide a review of studies that have discussed its impact on the dependent variable; however, these are explanations of a direct effect rather than a moderation effect. In some cases authors end the explanation with a couple of statements along the lines of “since the independent variable has an impact on the dependent variable and the moderating variable also has an impact on the dependent variable, one can conclude that the moderating variable interacts with the independent variable to affect the dependent variable, leading to a hypothesis that argues for the interaction”. However, such explanatory strategy does not actually provide an explanation of the interaction per se . A moderator or interaction variable may or may not have an effect on the dependent variable ( Carte & Russell, 2003 ); moreover, the independent and moderator variables should not be theoretically related as this would imply mediation ( Baron & Kenny, 1986 ). Thus, the arguments for a variable’s moderating effect on the main relationship must be distinct from its direct effect on the dependent variable, and if there is a relationship between the moderator and dependent variable, the underlying theoretical mechanism linking them must differ from the theoretical mechanism that influences the main relationship.

Third, some papers face the challenge that although they discuss the relationship between independent and moderating variables on the dependent variable, they do not actually explain the direction of the relationship in the interaction effect. This is problematic especially when the direction of causality can theoretically go both ways. As a result of not specifying theoretically the direction of the relationship, it is unclear which is the main mechanism and which is the interaction effect. A discussion of how the variables have been found to be related to each other in previous research, or indeed how the interaction among variables has been found to be statistically significant in previous studies, does not qualify as an explanation of causality. Most of the interaction effects are statistically analyzed by simply multiplying the independent and moderating variables and studying how this product term affects the dependent variable. While the empirical results are the same, the theoretical implications of the direction of causality are not equivalent, and conclusions and recommendations drawn from an inaccurate causality relationship may be erroneous.

Recommendations for Explaining Within-Level Interaction Effects

The typical figure explaining a within-level interaction effect appears in Figure 1 . This representation is commonly found in manuscripts in which the relationship between the independent variable of interest ( X ) and the dependent variable ( Y ) is argued to be modified by some other variable ( Z ). This simple representation then results in one or two hypotheses being discussed and formally presented in the paper. The first hypothesis tends to be the direct effect, predicting the impact of the independent variable of interest on the dependent variable. The second hypothesis is the moderation effect, predicting the strengthening or weakening of the direct effect under the moderating condition (see Cuervo-Cazurra & Dau, 2009 , as an illustration).

Typical relationships in a within-level moderation model.

Note : The thicker line is the relationship of focus.

Although parsimony can be useful, authors need to be careful and avoid oversimplifying to the point of making simplistic arguments. Figure 2 presents some of the potential additional relationships that may have an influence on the explanation of the moderating relationships and that need to be theoretically addressed: the direct effect of the moderating variable on the dependent variable, the reverse interaction (i.e., the independent variable becoming the moderating variable), and alternative explanations for the moderation effect. Although other relationships, such as mediation and additional exogenous variables, may also have an effect on the moderation, they are beyond the scope of this article.

Additional relationships in need of theoretical explanation when analyzing a within-level moderation model.

Note : Dashed lines indicate alternative relationships to consider.

We now provide a sequence of steps that authors can use to ensure that the interaction effects are clearly explained. In some cases, there is little empirical literature one can use to justify the proposed relationship. In such cases, it is even more important that the moderation effect is clearly explained and that the choice of moderating variables, as well as the proposed nature and effects of these on the direct relationship, is clearly guided by theory. This does not imply that the explanation should not include citations, but rather that the author needs to outline and articulate the underlying theoretical basis of the conceptual mechanisms to explain the interaction. We suggest the following steps to explain within-level moderation:

First, identify the theory or theories that are used to explain the direct and moderating effects. Clearly stating the theory used, the logic for using such theory, and an outline of the key arguments and assumptions of the theory not only helps the author clarify the theoretical approach used to build hypotheses, but also helps the reader understand how the author explains the arguments.

Second, apply the selected theory to the research question and explain the direct effect and the mechanisms behind it. The explanation of the mechanisms requires statement such as “variable x has a positive effect on variable y , because …”. This does not imply a discussion of how the variables are related, but rather one of why they are related and why the causality goes in a particular direction. If the direct effect has been widely analyzed before and there is a consensus on the relationship from the theoretical standpoint, you may state that this is a well-known argument, and that the direct effect is merely a baseline hypothesis.

Third, provide a theoretical justification for the choice of moderator variable. The inclusion of moderating effects in the analysis must be driven by theory rather than by the existence of previous empirical studies that have discussed such interaction, or by the statistical significance of the interaction term in the statistical analysis. The moderating variable establishes conditions under which the direct effect varies, and thus its selection needs to be within the realm of the theory used. Even if you find a statistically significant interaction effect, this does not mean that the moderating variable is theoretically justified; you may be finding such effect because there is mediation or because there is a common determinant (for a discussion, see Frazier, Tix, & Barron, 2004 ).

Fourth, explain the direct effect, if any, of the moderator variable on the dependent variable so that it is clear how this direct effect differs from the interaction effect. Although this may be a well-explained relationship, you still need to clarify the mechanisms that lead the moderating variable to affect the dependent variable. As with the direct effect, you may want to present a separate baseline hypothesis if it is relevant. These mechanisms need to differ not only from the mechanisms explaining the interaction effect, but also from the mechanisms explaining the direct effect of the independent variable on the dependent variable.

Fifth, explain how the interaction changes the mechanisms that explain the direct relationship. Using theory, specify arguments such as “the impact of X on Y is strengthened when Z is present because Z changes the mechanism in this manner …” or “the influence of X on Y is reduced in the presence of Z because the mechanism is weakened in this way …”. Conceptualized as a contingency hypothesis, moderation can be used to examine the boundaries and limitations of a theory ( Boyd, Haynes, Hitt, Bergh, & Ketchen, 2012 ). In this way, moderation specifies the conditions under which a given theory applies (or not) and thus increases the precision of theoretical predictions ( Edwards, 2010 ). Again, make sure that the explanation of the interaction effect differs from the explanation of the direct effect as well as from the explanation of the impact of the moderating variable on the dependent variable.

Sixth, theoretically rule out the reverse interaction in which the independent variable X is moderating the relationship between the moderating variable Z and the dependent variable Y . This of course only becomes an issue if a theoretical rationale exists for linking Z to Y , which we discussed in point four above. The theoretical challenge is to argue that the moderation can only exist in one direction and not the other, for example, because the moderator operates at a different level of analysis or temporally precedes the relationship. Phrase the hypotheses and graph the interactions in a way that is consistent with the theoretically grounded direction of the moderating relationship ( Aguinis, Gottfredson, & Culpepper, 2013 ).

Seventh, return to theory when interpreting the results and explain them from a theoretical viewpoint. Rather than state the usual “hypothesis x is supported because the coefficient of the interaction term is statistically significant”, put far more emphasis on the substantive meaning of such results in terms of our theoretical understanding of the phenomenon under investigation. Specify whether the nature and/or strength of the focal relationship changed as a result of the inclusion of the interaction and how such results inform theory and research moving forward. Non-significant results from the inclusion of moderation effects may also provide useful insights.

Recommendations for Explaining Cross-Level Interaction Effects

Multilevel studies involve relationships between independent and dependent variables at different levels; thus cross-level relationships can be direct and/or moderating. Applying multilevel lenses requires both conceptual and analytical considerations ( Snijders & Bosker, 2012 ).

The cross-country nature of IB is particularly ripe for multilevel studies and cross-level interactions ( Peterson, Arregle, & Martin, 2012 ). Multilevel theorizing provides ample opportunities for cross-fertilization of theories originating from different disciplines; at the same time it requires careful attention to the underlying assumptions of those theories. One typical use of cross-level interaction in IB is the analysis of the impact of country-level variables on firm-level behavior. A baseline hypothesis may be grounded in well-known IB theory, with the contribution to the literature coming in the form of a modification of the expected relationship based on insights from another theory that operates at a different level.

However, simply adding another moderating variable, even at a different level of theory, does not constitute a theoretical contribution per se . While borrowing concepts and variables from different disciplines may yield new insights, an in-depth appreciation for the underlying theory and rigorous integration with IB theory is paramount ( Bello & Kostova, 2012 ; Cheng, Henisz, & Roth, 2009 ). It is critical to avoid committing the error of “rebottling old wine in new bottles”; that is, selecting variables that have previously been studied in similar settings. Variables and concepts from other levels or disciplines are often part of a system of constructs that together make up a theory; separating (i.e., cherry picking) one or two of these constructs and utilizing them in an isolated fashion as moderators in IB studies violate the underlying coherence of the theory and constructs and may lead to flawed theorizing. Moreover, though drawing on concepts from theories at different levels holds much promise for advancing IB theory, and we certainly promote such theorizing, careful attention must be paid to the ability of theoretical constructs to traverse levels without losing their substantive meaning. Key constructs may be subject to different meanings at different levels, in different cultures and environments, or between headquarters and subsidiaries.

The prevailing logic in management is that the larger context within which organizations are embedded exerts a greater downward influence than vice versa ( Mathieu & Chen, 2011 ). In cross-level research, accurately accounting for the nesting structure is critical because observations within higher level units are more similar than observations across those units; since lower level units share common characteristics and influences from the higher level units, they are not independent from each other. For instance, in IB, subsidiaries are nested within the multinational firm, which is in turn hierarchically nested within its home country. Lower level units may, however, belong simultaneously to multiple higher levels (i.e., industries and countries). To the extent that industries are not country specific but rather global in nature, the resulting structure is non-hierarchical or cross-classified, since each firm uniquely belongs to a combination of both home country and industry levels (for a discussion of different nesting structures in IB research, see Nielsen & Nielsen, 2010 ). In IB, upward cross-level influences can be theorized (e.g., multinational firms influencing a host-country institutional environment via lobbying and non-market strategies); however, such influences are typically main rather than moderating effects. Clear specification of the levels of theory and variables helps conceptualize the nature and direction of cross-level relationships.

Figure 3 depicts a typical hierarchical nesting structure with three levels (e.g., subsidiaries within the multinational firm, nested within the home country institutional environment), allowing for simple Level 3–1 and Level 2–1 cross-level interactions.

Typical relationships in a cross-level moderation model.

Note : The thicker lines are the relationships of focus.

Figure 3 is, however, an oversimplification of the reality surrounding cross-level interaction, as several additional relationships must be recognized and discussed. First, Levels 2 and 3 moderator variables ( Z, W ) may also exert direct (downward) influence on the dependent variable ( Y ). Such potential influences must be acknowledged and accounted for both theoretically and empirically. Moreover, the potential for reverse or symmetrical interaction effects should be ruled out. Multilevel modeling can help identify the directionality of the interaction effects in that it is logical that the contextual variable moderates the relationship between lower level variables ( Aguinis et al., 2013 ). However, the theoretical rationale for directionality of interaction effects must still be specified, with constructs and measurement treated accordingly ( Klein, Dansereau, & Hall, 1994 ). Multilevel studies of this kind can develop and test hypotheses pertaining to three types of relationships: (a) lower level direct effects (Level 1–1); (b) cross-level direct effects (Levels 2–1 and 3–1); and (3) cross-level interaction effects (Levels 2–1 and 3–1). Manuscripts with cross-level interactions tend to discuss the direct effect first followed by the moderation of this effect by Levels 2 and 3 cross-level interactions. Figure 4 illustrates these additional relationships associated with cross-level interactions.

Additional relationship in need of theoretical explanation when analyzing a cross-level moderation model.

Similar to the previous discussion, we offer a set of concrete steps on how to develop theoretical insights for authors considering cross-level interactions:

First, specify the focal unit of analysis of the study. This is typically determined by the analytical level of the dependent variable (e.g., firm for studies analyzing firm performance; team for studies analyzing innovation in teams) and represents the level to which generalizations are made. This first step is important, as the unit of analysis determines the appropriate level of associated theoretical constructs and helps avoid misattribution of effects, commonly referred to as “fallacies of the wrong level” ( Rousseau, 1985 : 5).

Second, specify the hierarchy and nature of theoretical nesting (e.g., individuals, teams, firm, industry, country, region, etc.). Nesting is important because lower level units share commonalities with higher level units. You need to determine the appropriate levels at which your phenomenon is operating and where you plan to draw the boundaries of the theoretical extension. For example, you may be interested in analyzing subsidiaries, nested within a multinational firm that is in turn headquartered in a particular country. In such a three-level hierarchical model, the multinational firm’s headquarter and/or home country characteristics may act as moderators on the relationship between subsidiary characteristics and strategic choice (e.g., Goerzen, Asmussen, & Nielsen, 2013 ).

Third, choose relevant independent variables from theories at each level of nesting and clearly specify their relationships with the dependent variable (upward or downward). In the selection of variables, take into account how the theory to which they adhere, or its extension, operates at the level of analysis of the focal unit. Explicit integration of theories that span different levels holds great potential for facilitating new theory generation; however, careful attention must be paid to how theoretical constructs operate across levels without losing their substantive meaning. For instance, trust at the institutional level may mean something different than inter-organizational or inter-personal trust ( Nielsen, 2010 ).

Fourth, model the within-group variance by specifying the lower level (Level 1–1) direct effects. Identify the theoretical mechanisms explaining these effects and make arguments consistent with the level of theory (i.e., the dependent variable). Direct effects hypotheses at the lower level are typically specified in the same manner as regular hypotheses, using statements like “predictor X is positively/negatively associated with outcome Y ”, or “the influence of predictor X on outcome Y is positive/negative”.

Fifth, choose relevant moderator variables from theories at higher (lower) levels. If the moderator variable has a relationship with the dependent variable, clearly distinguish the theoretical arguments for the main effect from those for the moderating effect.

Sixth, model the between-group variance in intercepts by specifying cross-level (Levels 2–1 and 3–1) direct effects. Identify the theoretical mechanisms explaining these effects and make arguments consistent with downward (upward) direct influences. Cross-level direct effects hypotheses are often specified using statements like: “industry competition negatively influences firm performance”, or “host country governance quality is positively associated with non-equity entry mode”.

Seventh, model the between-group variance in slopes by specifying cross-level (Levels 2–1 and 3–1) interactions. Identify the theoretical mechanisms that explain how and why the nature or strength of the lowest level relationships changes as a function of the higher level moderator. Cross-level interaction effects hypotheses are often specified using statements like “the relationship between firm international diversification and performance varies with home country institutional quality such that firms originating from countries with higher quality institutional environments are more likely to benefit from international diversification than firms originating from countries with lower quality institutional environments”.

Eighth, rule out reverse interaction of the independent variable on the cross-level direct relationship between the moderator and the dependent variable. The statistical analysis software cannot detect the direction of relationships and theory must guide this choice. Cross-level interactions typically involve contextual variables at higher levels and this often makes it easier to rule out reverse interaction from a logical standpoint; it is far more likely that industry- or country-level factors moderate the relationship between firm strategy and performance than it is that direct effects of industry or country characteristics vary with one firm’s strategy or conduct. Depicting the research model in a figure helps clarify theoretical nesting and the nature and direction of relationships and interactions.

Ninth, return to theory when interpreting the results. Multilevel research offers the opportunity to extend theory by bridging or integrating theories from different domains. You need to explain how the cross-level effects (direct or interaction) change our understanding of the theoretical mechanisms that link concepts in a model. It is often useful to examine to what extent the cross-level interactions modify both the theory of the focal unit of analysis and theories at higher levels from which the moderators are drawn.

CONCLUSIONS

Interaction effects are increasingly being analyzed in research papers. This is especially the case in IB, because the cross-disciplinary nature of the phenomenon enables researchers to generate new insights by analyzing the boundary conditions of well-known relationships that have hitherto been explained in a domestic setting. Given the theoretical challenges posed by interaction effects, in this editorial we provided a sequence of steps that researchers can follow to explain interaction effects within and across levels of analysis. These steps should be viewed as tools that can be adapted and modified depending on the specific research question and nature of data, rather than strict steps that all submitted papers must follow. The objective of these suggestions is to create papers that provide deeper discussions and extensions of theory. This editorial complements other editorials that have discussed how to develop theory in IB ( Bello & Kostova, 2012 ; Cheng et al., 2009 ; Cuervo-Cazurra, Caligiuri, Andersson, & Brannen, 2013 ; Thomas, Cuervo-Cazurra, & Brannen, 2011 ) and how to incorporate advanced statistical techniques ( Chang, van Witteloostuijn, & Eden, 2010 ; Peterson et al., 2012 ; Reeb, Sakakibara, & Mahmood, 2012 ).

We use the term mechanisms to denote underlying theoretical processes (or reasons) for certain proposed effects. This is different from the use of mechanisms to denote intervening (mediating) variables in a causal chain of relationships (see Baron & Kenny, 1986 ).

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Acknowledgements

We thank Editor John Cantwell and two anonymous reviewers for useful suggestions for improvement. Alvaro Cuervo-Cazurra thanks the Patrick F. and Helen C. Walsh Research Professorship and the Robert Morrison Research Fellowship at Northeastern University for financial support. All errors are ours.

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Andersson, U., Cuervo-Cazurra, A. & Nielsen, B. From the Editors: Explaining interaction effects within and across levels of analysis. J Int Bus Stud 45 , 1063–1071 (2014). https://doi.org/10.1057/jibs.2014.50

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Published : 05 November 2014

Issue Date : 01 December 2014

DOI : https://doi.org/10.1057/jibs.2014.50

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13.2.1: Example with Main Effects and Interactions

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• Page ID 22132

• Michelle Oja
• Taft College

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You might realize by now the Dr. MO’s student researchers did a couple studies on mindset. Remember growth mindset ? The idea that mistakes help you learn, and that more time and effort leads to brain development? One group of student researchers collected data on improving growth mindset that should be analyzed with a factorial ANOVA. Here it is:

The student researchers looked at the Difference scores of 106 students at Dr. MO’s community college. The Difference scores were calculated by subtracting each students’ Post-Test Mindset Quiz score (measured at the end of the semester) from their own Pre-Test Mindset Quiz score (measured at the beginning of the same semester); thus, positive scores mean that the student improved their mindset. In this study, we had two variables. One was the intervention, which had two levels: No Intervention (comparison control group) and Intervention (in which faculty tried different activities to improve mindset). We also collected information on what department these activities were happening in, and had enough data to analyze students in English classes and students in Psychology classes.

Can you identify the groups and variables in this scenario?

Example \(\PageIndex{1}\)

Answer the following questions to understand the variables and groups that we are working with.

• Who is the sample?
• Who do might be the population?
• What are the IVs and levels for each IV?
• What is the DV (quantitative variable being measured)?
• Is this a 2x2 factorial design? If not, what kind of design is it?
• List out each of the combinations of the levels of the IVs:
• The sample was 106 community college students taking an English class or a Psychology class.
• The population could be all community colleges students, or maybe all community college students taking English or Psychology. Or maybe all community college students in a general education course?
• One IV is Intervention, with the levels being Yes or No. The other IV is Department, with the levels being English or Psychology.
• The DV is the Difference score from the Mindset Quiz pre-test to the Mindset Quiz post-test.
• Yes, this is a 2x2 factorial design because there are two IVs (so there are two numbers __x__); the first IV (Intervention) has two levels and the second IV (Department) has two levels.
• Intervention in English
• Intervention in Psychology
• No Intervention in English
• No Intervention in Psychology

Can you plug these IVs into a Punnett’s Square grid?

Example \(\PageIndex{2}\)

Complete a grid showing the factorial design IVs and DVS.

You could have put Department as IV1 (columns) and Intervention as IV2 (rows). It’s really about what makes the most sense to you; Dr. MO wanted what she thinks is the main influence of mindset on the top.

Participants in each “cell” of this design have a unique combination of IV conditions.

Three Effects

With a 2x2 factorial design, you have three effects to look at. Remember, “effects” are the results of the DV, what was measured. Here are the three effects that you need to look at:

• The main effect of the one IV: How does one IV affect the DV (independent of the other IV)
• The main effect of the other IV: How does the other IV affect the DV (independent of the first IV)
• The interaction of the two IVs -- how they jointly affect the DV

Example \(\PageIndex{3}\)

In our mindset scenario, what are these three effects?

• The main effect of the intervention ; did the intervention improve Mindset Quiz scores?
• The main effect of the department ; did which class you were in affect Mindset Quiz scores?
• The interaction of the intervention by department ; is one department more likely to improve Mindset Quiz scores than another department when there’s an intervention? Another way to think about these variables is to ask whether the departments started out with different average Mindset Quiz scores, so that even in the No Intervention condition the Difference score would be statistically significantly different between the two departments.

Let’s look at these main effects in Table \(\PageIndex{2}\), in which the marginal means were included. Marginal means are, you guessed, it the means on the margins of the table. These means on the margin show the means for each level of each IV, which are the main effects . The marginal means do not show the combination of the IVs’ levels, so they do not show an interaction.

Again, you could have put Department as IV1 (columns) and Intervention as IV2 (rows). And again, participants in each “cell” of this design have a unique combination of IV conditions.

Main Effects

Let’s go through the marginal means for Table \(\PageIndex{2}\).

What are the marginal means for the Intervention?

Intervention Yes \(=\) 2.08

Intervention No \(=\)1.76

We’ll look at statistical significant later, but just based on these means, it looks like the students who experienced an Intervention had a larger Difference score than students who did not have an intervention. This is what was expected; students who experienced an intervention had a higher Mindset Quiz score at the end of the semester (post-test), after the interventions, than at the beginning of the semester (pretest). This is your main effect of Intervention. When we do the statistical analyses, we’ll follow the same process for null hypothesis significance testing:

Critical \(< |Calculated| = \) Reject null \(=\) means are different \(=\) main effects \(= p<.05\)

Critical \(> |Calculated| =\) Retain null \(=\) means are similar \(=\) no main effects \(= p>.05\)

Let’s move on to the other independent variable.

Exercise \(\PageIndex{1}\)

Based on Table \(\PageIndex{2}\), what are the marginal means for department? Which department seemed to have a higher Difference score?

Marginal Means:

• English=2.49
• Psychology=1.34

There seems to be a main effect of department such that students in English had a higher Difference score than students in Psychology classes.

We can’t be say much more than that without looking at actual statistical results. Instead, we will look at the individual cells of our grid to see if there was an interaction between Department and Intervention on the Difference scores.

Interaction

Table \(\PageIndex{3}\) has the means for each combination of each IV level in the individual cells. Looking at Table \(\PageIndex{3}\), how do the inside cells seem relate to each other?

Example \(\PageIndex{5}\)

Answer the following questions related the cell means in Table \(\PageIndex{3}\),

• Was one cell substantially higher than the others?
• Was one cell substantially lower than the others?
• It looks like students in English class who experienced an intervention had higher Difference scores.
• It looks like students in a Psychology class who experienced an intervention had a really low difference score.

This pattern of cell means is your interaction of Intervention and Department: Experiencing the intervention interacts with the department to affect Differences in Mindset Quiz scores such that the intervention seems to improve mindset for those in English but not in Psychology (the Psychology students who experienced an intervention were essentially unchanged). What’s strange is that Psychology students in the control condition (No Intervention) might have actually improved their Mindset Quiz scores more than both English students with no intervention and Psychology student with the intervention! You might be wondering why these strange effects for the Psychology students, and Dr. MO has no answer for you. This is real data, and there’s no good explanation for this with the variables that we have. ::shrug::

As you can see, the main effects of each IV can relate to the interaction in several different ways. Let’s look at that next.

Research Hypothesis In Psychology: Types, & Examples

Saul Mcleod, PhD

Editor-in-Chief for Simply Psychology

BSc (Hons) Psychology, MRes, PhD, University of Manchester

Saul Mcleod, PhD., is a qualified psychology teacher with over 18 years of experience in further and higher education. He has been published in peer-reviewed journals, including the Journal of Clinical Psychology.

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A research hypothesis, in its plural form “hypotheses,” is a specific, testable prediction about the anticipated results of a study, established at its outset. It is a key component of the scientific method .

Hypotheses connect theory to data and guide the research process towards expanding scientific understanding

Some key points about hypotheses:

• A hypothesis expresses an expected pattern or relationship. It connects the variables under investigation.
• It is stated in clear, precise terms before any data collection or analysis occurs. This makes the hypothesis testable.
• A hypothesis must be falsifiable. It should be possible, even if unlikely in practice, to collect data that disconfirms rather than supports the hypothesis.
• Hypotheses guide research. Scientists design studies to explicitly evaluate hypotheses about how nature works.
• For a hypothesis to be valid, it must be testable against empirical evidence. The evidence can then confirm or disprove the testable predictions.
• Hypotheses are informed by background knowledge and observation, but go beyond what is already known to propose an explanation of how or why something occurs.

Predictions typically arise from a thorough knowledge of the research literature, curiosity about real-world problems or implications, and integrating this to advance theory. They build on existing literature while providing new insight.

Types of Research Hypotheses

Alternative hypothesis.

The research hypothesis is often called the alternative or experimental hypothesis in experimental research.

It typically suggests a potential relationship between two key variables: the independent variable, which the researcher manipulates, and the dependent variable, which is measured based on those changes.

The alternative hypothesis states a relationship exists between the two variables being studied (one variable affects the other).

A hypothesis is a testable statement or prediction about the relationship between two or more variables. It is a key component of the scientific method. Some key points about hypotheses:

• Important hypotheses lead to predictions that can be tested empirically. The evidence can then confirm or disprove the testable predictions.

In summary, a hypothesis is a precise, testable statement of what researchers expect to happen in a study and why. Hypotheses connect theory to data and guide the research process towards expanding scientific understanding.

An experimental hypothesis predicts what change(s) will occur in the dependent variable when the independent variable is manipulated.

It states that the results are not due to chance and are significant in supporting the theory being investigated.

The alternative hypothesis can be directional, indicating a specific direction of the effect, or non-directional, suggesting a difference without specifying its nature. It’s what researchers aim to support or demonstrate through their study.

Null Hypothesis

The null hypothesis states no relationship exists between the two variables being studied (one variable does not affect the other). There will be no changes in the dependent variable due to manipulating the independent variable.

It states results are due to chance and are not significant in supporting the idea being investigated.

The null hypothesis, positing no effect or relationship, is a foundational contrast to the research hypothesis in scientific inquiry. It establishes a baseline for statistical testing, promoting objectivity by initiating research from a neutral stance.

Many statistical methods are tailored to test the null hypothesis, determining the likelihood of observed results if no true effect exists.

This dual-hypothesis approach provides clarity, ensuring that research intentions are explicit, and fosters consistency across scientific studies, enhancing the standardization and interpretability of research outcomes.

Nondirectional Hypothesis

A non-directional hypothesis, also known as a two-tailed hypothesis, predicts that there is a difference or relationship between two variables but does not specify the direction of this relationship.

It merely indicates that a change or effect will occur without predicting which group will have higher or lower values.

For example, “There is a difference in performance between Group A and Group B” is a non-directional hypothesis.

Directional Hypothesis

A directional (one-tailed) hypothesis predicts the nature of the effect of the independent variable on the dependent variable. It predicts in which direction the change will take place. (i.e., greater, smaller, less, more)

It specifies whether one variable is greater, lesser, or different from another, rather than just indicating that there’s a difference without specifying its nature.

For example, “Exercise increases weight loss” is a directional hypothesis.

Falsifiability

The Falsification Principle, proposed by Karl Popper , is a way of demarcating science from non-science. It suggests that for a theory or hypothesis to be considered scientific, it must be testable and irrefutable.

Falsifiability emphasizes that scientific claims shouldn’t just be confirmable but should also have the potential to be proven wrong.

It means that there should exist some potential evidence or experiment that could prove the proposition false.

However many confirming instances exist for a theory, it only takes one counter observation to falsify it. For example, the hypothesis that “all swans are white,” can be falsified by observing a black swan.

For Popper, science should attempt to disprove a theory rather than attempt to continually provide evidence to support a research hypothesis.

Can a Hypothesis be Proven?

Hypotheses make probabilistic predictions. They state the expected outcome if a particular relationship exists. However, a study result supporting a hypothesis does not definitively prove it is true.

All studies have limitations. There may be unknown confounding factors or issues that limit the certainty of conclusions. Additional studies may yield different results.

In science, hypotheses can realistically only be supported with some degree of confidence, not proven. The process of science is to incrementally accumulate evidence for and against hypothesized relationships in an ongoing pursuit of better models and explanations that best fit the empirical data. But hypotheses remain open to revision and rejection if that is where the evidence leads.

• Disproving a hypothesis is definitive. Solid disconfirmatory evidence will falsify a hypothesis and require altering or discarding it based on the evidence.
• However, confirming evidence is always open to revision. Other explanations may account for the same results, and additional or contradictory evidence may emerge over time.

We can never 100% prove the alternative hypothesis. Instead, we see if we can disprove, or reject the null hypothesis.

If we reject the null hypothesis, this doesn’t mean that our alternative hypothesis is correct but does support the alternative/experimental hypothesis.

Upon analysis of the results, an alternative hypothesis can be rejected or supported, but it can never be proven to be correct. We must avoid any reference to results proving a theory as this implies 100% certainty, and there is always a chance that evidence may exist which could refute a theory.

How to Write a Hypothesis

• Identify variables . The researcher manipulates the independent variable and the dependent variable is the measured outcome.
• Operationalized the variables being investigated . Operationalization of a hypothesis refers to the process of making the variables physically measurable or testable, e.g. if you are about to study aggression, you might count the number of punches given by participants.
• Decide on a direction for your prediction . If there is evidence in the literature to support a specific effect of the independent variable on the dependent variable, write a directional (one-tailed) hypothesis. If there are limited or ambiguous findings in the literature regarding the effect of the independent variable on the dependent variable, write a non-directional (two-tailed) hypothesis.
• Make it Testable : Ensure your hypothesis can be tested through experimentation or observation. It should be possible to prove it false (principle of falsifiability).
• Clear & concise language . A strong hypothesis is concise (typically one to two sentences long), and formulated using clear and straightforward language, ensuring it’s easily understood and testable.

Consider a hypothesis many teachers might subscribe to: students work better on Monday morning than on Friday afternoon (IV=Day, DV= Standard of work).

Now, if we decide to study this by giving the same group of students a lesson on a Monday morning and a Friday afternoon and then measuring their immediate recall of the material covered in each session, we would end up with the following:

• The alternative hypothesis states that students will recall significantly more information on a Monday morning than on a Friday afternoon.
• The null hypothesis states that there will be no significant difference in the amount recalled on a Monday morning compared to a Friday afternoon. Any difference will be due to chance or confounding factors.

More Examples

• Memory : Participants exposed to classical music during study sessions will recall more items from a list than those who studied in silence.
• Social Psychology : Individuals who frequently engage in social media use will report higher levels of perceived social isolation compared to those who use it infrequently.
• Developmental Psychology : Children who engage in regular imaginative play have better problem-solving skills than those who don’t.
• Clinical Psychology : Cognitive-behavioral therapy will be more effective in reducing symptoms of anxiety over a 6-month period compared to traditional talk therapy.
• Cognitive Psychology : Individuals who multitask between various electronic devices will have shorter attention spans on focused tasks than those who single-task.
• Health Psychology : Patients who practice mindfulness meditation will experience lower levels of chronic pain compared to those who don’t meditate.
• Organizational Psychology : Employees in open-plan offices will report higher levels of stress than those in private offices.
• Behavioral Psychology : Rats rewarded with food after pressing a lever will press it more frequently than rats who receive no reward.

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Section 7.3: Moderation Models, Assumptions, Interpretation, and Write Up

Learning Objectives

At the end of this section you should be able to answer the following questions:

• What are some basic assumptions behind moderation?
• What are the key components of a write up of moderation analysis?

Moderation Models 

Difference between mediation & moderation.

The main difference between a simple interaction, like in ANOVA models or in moderation models, is that mediation implies that there is a causal sequence. In this case, we know that stress causes ill effects on health, so that would be the causal factor.

Some predictor variables interact in a sequence, rather than impacting the outcome variable singly or as a group (like regression).

Moderation and mediation is a form of regression that allows researchers to analyse how a third variable effects the relationship of the predictor and outcome variable.

Moderation analyses imply an interaction on the different levels of M

PowerPoint: Basic Moderation Model

Consider the below model:

• Chapter Seven – Basic Moderation Model

Would the muscle percentage be the same for young, middle-aged, and older participants after training? We know that it is harder to build muscle as we age, so would training have a lower effect on muscle growth in older people?

Example Research Question:

Does cyberbullying moderate the relationship between perceived stress and mental distress?

Moderation Assumptions

• The dependent and independent variables should be measured on a continuous scale.
• There should be a moderator variable that is a nominal variable with at least two groups.
• The variables of interest (the dependent variable and the independent and moderator variables) should have a linear relationship, which you can check with a scatterplot.
• The data must not show multicollinearity (see Multiple Regression).
• There should be no significant outliers, and the distribution of the variables should be approximately normal.

Moderation Interpretation

PowerPoint: Moderation menu, results and output

Please have a look at the following link for the Moderation Menu and Output:

• Chapter Seven – Moderation Output

Interpretation

The effects of cyberbullying can be seen in blue, with the perceived stress in green. These are the main effects of the X and M variable on the outcome variable (Y). The interaction effect can be seen in purple. This will tell us if perceived stress is effecting mental distress equally for average, lower than average or higher than average levels of cyberbullying. If this is significant, then there is a difference in that effect. As can be seen in yellow and grey, cyberbullying has an effect on mental distress, but the effect is stronger for those who report higher levels of cyberbullying (see graph).

Moderation Write Up

The following text represents a moderation write up:

A moderation test was run, with perceived stress as the predictor, mental distress as the dependant, and cyberbullying as a moderator.  There was a significant main effect found between perceived stress and mental distress, b = -1.23, BCa CI [1.11, 1.34], z =21.38 , p <.001, and nonsignificant main effect of cyberbullying on mental distress b = 1.05, BCa CI [0.72, 1.38], z=6.28, p < .001. There was a significant interaction found by cyberbullying on perceived stress and mental distress, b = -0.05, BCa CI [0.01, 0.09], z=2.16, p =.031. It was found that participants who reported higher than average levels of cyberbullying experienced a greater effect of perceived stress on mental distress ( b = 1.35, BCa CI [1.19, 1.50], z=17.1, p < .001), when compared to average or lower than average levels of cyberbullying ( b = 1.23, BCa CI [1.11, 1.34], z=21.3, p < .001, b = 1.11, BCa CI [0.95, 1.27], z=13.8, p < .001, respectively). From these results, it can be concluded that the effect of perceived stress on mental distress is partially moderated by cyberbullying.

Statistics for Research Students Copyright © 2022 by University of Southern Queensland is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

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The Interaction Hypothesis is a type of theory proposing that one of most effective methods of learning a new language is through personal and direct interaction. This theory is applied specifically to the acquisition of a foreign or a second language . It is usually attributed to Professor Michael Long, when he wrote a paper entitled “The Role of the Linguistic Environment in Second Language Acquisition ” in 1996.

Through the Interaction Hypothesis, Professor Long integrated and reconciled two hypotheses on second language acquisition (SLA): the input and the output hypotheses. The Input Hypothesis states that a language learner only needs to be supplied with “input” through the forms of reading, listening to conversations, and lessons on grammar and vocabulary. The Output Hypothesis, on the other hand, stresses the importance of practicing and speaking to retain and remember the language. The Interaction Hypothesis combines both the “input” and “output” by stating that interaction is not only a means for a learner to study the language, but also a way for the learner to practice what he has learned.

Among the types of interactions, conversation is probably the most emphasized in the Interaction Hypothesis, an idea most probably derived from the “discourse approach” by Professor Evelyn Hatch who, in 1978, wrote papers that stressed the importance of constant communication and interaction for SLA. The Interaction Hypothesis acknowledges that during conversations, there are certain situations wherein a participant does not understand what the other says, but it is in these situations where learning becomes more effective. The theory refers to this occurrence as “negotiation,” wherein the participants will attempt to understand and repair the miscommunication during the interaction.

The first step in the negotiation is the interaction itself, when both participants begin to engage in conversation. The second step, the “negative feedback,” occurs when a participant does not understand a certain word, sometimes seen in a nonverbal action such as in the furrowing of the brow. In some cases, the other participant may request clarification by saying, “Pardon?” or “Can you say that again?” The process wherein the misunderstood participant strives to make the other participant understand is called “modification output.” The participant may paraphrase or give examples to make the meaning of the word clearer, until the other participant responds in an affirmative way that he has understood.

Interaction Hypothesis suggests an interaction between a second-language learner and a native speaker, so the learner can study the language in its most authentic setting. In this way, the learner not only learns about the language, but also the nuances and other nonverbal cues the go along with the words. Many universities in English-speaking countries have English programs and classes focusing on personal interaction for many foreign students who go abroad just to learn how to speak English.

Related Articles

• What Is Conversation Theory?
• What Is the Input Hypothesis?
• What Are the Different Types of Second Language Acquisition Theories?
• What is the Social Exchange Theory?

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Statistics Made Easy

Understanding the Null Hypothesis for ANOVA Models

A one-way ANOVA is used to determine if there is a statistically significant difference between the mean of three or more independent groups.

A one-way ANOVA uses the following null and alternative hypotheses:

• H 0 :  μ 1  = μ 2  = μ 3  = … = μ k  (all of the group means are equal)
• H A : At least one group mean is different   from the rest

To decide if we should reject or fail to reject the null hypothesis, we must refer to the p-value in the output of the ANOVA table.

If the p-value is less than some significance level (e.g. 0.05) then we can reject the null hypothesis and conclude that not all group means are equal.

A two-way ANOVA is used to determine whether or not there is a statistically significant difference between the means of three or more independent groups that have been split on two variables (sometimes called “factors”).

A two-way ANOVA tests three null hypotheses at the same time:

• All group means are equal at each level of the first variable
• All group means are equal at each level of the second variable
• There is no interaction effect between the two variables

To decide if we should reject or fail to reject each null hypothesis, we must refer to the p-values in the output of the two-way ANOVA table.

The following examples show how to decide to reject or fail to reject the null hypothesis in both a one-way ANOVA and two-way ANOVA.

Example 1: One-Way ANOVA

Suppose we want to know whether or not three different exam prep programs lead to different mean scores on a certain exam. To test this, we recruit 30 students to participate in a study and split them into three groups.

The students in each group are randomly assigned to use one of the three exam prep programs for the next three weeks to prepare for an exam. At the end of the three weeks, all of the students take the same exam. 

The exam scores for each group are shown below:

When we enter these values into the One-Way ANOVA Calculator , we receive the following ANOVA table as the output:

Notice that the p-value is 0.11385 .

For this particular example, we would use the following null and alternative hypotheses:

• H 0 :  μ 1  = μ 2  = μ 3 (the mean exam score for each group is equal)

Since the p-value from the ANOVA table is not less than 0.05, we fail to reject the null hypothesis.

This means we don’t have sufficient evidence to say that there is a statistically significant difference between the mean exam scores of the three groups.

Example 2: Two-Way ANOVA

Suppose a botanist wants to know whether or not plant growth is influenced by sunlight exposure and watering frequency.

She plants 40 seeds and lets them grow for two months under different conditions for sunlight exposure and watering frequency. After two months, she records the height of each plant. The results are shown below:

In the table above, we see that there were five plants grown under each combination of conditions.

For example, there were five plants grown with daily watering and no sunlight and their heights after two months were 4.8 inches, 4.4 inches, 3.2 inches, 3.9 inches, and 4.4 inches:

She performs a two-way ANOVA in Excel and ends up with the following output:

We can see the following p-values in the output of the two-way ANOVA table:

• The p-value for watering frequency is 0.975975 . This is not statistically significant at a significance level of 0.05.
• The p-value for sunlight exposure is 3.9E-8 (0.000000039) . This is statistically significant at a significance level of 0.05.
• The p-value for the interaction between watering  frequency and sunlight exposure is 0.310898 . This is not statistically significant at a significance level of 0.05.

These results indicate that sunlight exposure is the only factor that has a statistically significant effect on plant height.

And because there is no interaction effect, the effect of sunlight exposure is consistent across each level of watering frequency.

That is, whether a plant is watered daily or weekly has no impact on how sunlight exposure affects a plant.

Additional Resources

The following tutorials provide additional information about ANOVA models:

How to Interpret the F-Value and P-Value in ANOVA How to Calculate Sum of Squares in ANOVA What Does a High F Value Mean in ANOVA?

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2 Replies to “Understanding the Null Hypothesis for ANOVA Models”

Hi, I’m a student at Stellenbosch University majoring in Conservation Ecology and Entomology and we are currently busy doing stats. I am still at a very entry level of stats understanding, so pages like these are of huge help. I wanted to ask, why is the sum of squares (treatment) for the one way ANOVA so high? I calculated it by hand and got a much lower number, could you please help point out if and where I went wrong?

As I understand it, SSB (treatment) is calculated by finding the mean of each group and the grand mean, and then calculating the sum of squares like this: GM = 85.5 x1 = 83.4 x2 = 89.3 x3 = 84.7

SSB = (85.5 – 83.4)^2 + (85.5 – 89.3)^2 + (85.5 – 84.7)^2 = 18.65 DF = 2

I would appreciate any help, thank you so much!

Hi Theo…Certainly! Here are the equations rewritten as they would be typed in Python:

### Sum of Squares Between Groups (SSB)

In a one-way ANOVA, the sum of squares between groups (SSB) measures the variation due to the interaction between the groups. It is calculated as follows:

1. **Calculate the group means**: “`python mean_group1 = 83.4 mean_group2 = 89.3 mean_group3 = 84.7 “`

2. **Calculate the grand mean**: “`python grand_mean = 85.5 “`

3. **Calculate the sum of squares between groups (SSB)**: Assuming each group has `n` observations: “`python n = 10 # Number of observations in each group

ssb = n * ((mean_group1 – grand_mean)**2 + (mean_group2 – grand_mean)**2 + (mean_group3 – grand_mean)**2) “`

### Example Calculation

For simplicity, let’s assume each group has 10 observations: “`python n = 10

ssb = n * ((83.4 – 85.5)**2 + (89.3 – 85.5)**2 + (84.7 – 85.5)**2) “`

Now calculate each term: “`python term1 = (83.4 – 85.5)**2 # term1 = (-2.1)**2 = 4.41 term2 = (89.3 – 85.5)**2 # term2 = (3.8)**2 = 14.44 term3 = (84.7 – 85.5)**2 # term3 = (-0.8)**2 = 0.64 “`

Sum these squared differences: “`python sum_of_squared_diffs = term1 + term2 + term3 # sum_of_squared_diffs = 4.41 + 14.44 + 0.64 = 19.49 ssb = n * sum_of_squared_diffs # ssb = 10 * 19.49 = 194.9 “`

So, the sum of squares between groups (SSB) is 194.9, assuming each group has 10 observations.

### Degrees of Freedom (DF)

The degrees of freedom for SSB is calculated as: “`python df_between = k – 1 “` where `k` is the number of groups.

For three groups: “`python k = 3 df_between = k – 1 # df_between = 3 – 1 = 2 “`

### Summary

– **SSB** should consider the number of observations in each group. – **DF** is the number of groups minus one.

By ensuring you include the number of observations per group in your SSB calculation, you can get the correct SSB value.

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Multilingual Pedagogy and World Englishes

Linguistic Variety, Global Society

Interaction Hypothesis

Michael Long writes,

“Whereas Krashen views comprehensible input (CI) one step ahead of the learner’s current level as necessary and sufficient for acquisition, I have long argued for the interaction hypothesis ….I maintain that CI is necessary but not sufficient for SLA…I have further argued for the importance of negotiation for meaning and negative feedback in orienting learners’ attention to form in this way” (788).

As a result, Long’s  interaction hypothesis, which does not refute but rather fills in perceived gaps in Krashen’s Input Hypothesis, suggests that comprehensible input is important, but the negotiations created by interactions between speaker and audience are an essential component in promoting language acquisition: “Modifications to the interactional structure of conversations which take place in the process of negotiating a communication problem help to make input comprehensible to an L2 learner” (Ellis, “The Interaction Hypothesis,” 4). Ellis also critiques aspects of the hypothesis, stating that while interactive negotiation and feedback can assist the learner,

“Sometimes interaction can overload learners with input, as when a speaker provides lengthy paraphrases or long definitions of unknown words. In such cases, acquisition may be impeded rather than facilitated” (Ellis, SLA , 48).

Application

The interaction hypothesis is one of many potential approaches to language learning pedagogy, but it has a lot of benefits in application. Interactivity in the classroom is not simply a good idea for promoting language acquisition; it also promotes a healthy, collaborative, and student-centered culture in which students will look to each other, in addition to their instructor, for assistance.

Bibliography

Ellis, Rod. “The Interaction Hypothesis: A Critical Evaluation.” ERIC, 1991, https://files.eric.ed.gov/fulltext/ED338037.pdf. Accessed 14 Apr. 2019.

—-.  Second Language Acquisition . Oxford, 1997.

Long, Michael H. “Two Commentaries on Ron Sheen’s ‘A Critical Analysis of the Advocacy of the Task-Based Syllabus’: On the Advocacy of the Task-Based Syllabus.”  TESOL Quarterly , vol. 28, no. 4, 1994, pp. 782–790.  JSTOR , www.jstor.org/stable/3587562.