Week 4 ANCOVA

How can you increase the F statistic value?

reduce the denominator, by conceptually accounting for variance in the DV that comes from other sources such as covariates,

It does this by using the ? of the relationship between the DV and the covariate as a reference point to calculate ? in order to improve the sensitivity of the analysis and hence increase the chances of a ?

regression slope, error, significant result.

So how many Covariates should you use in your model, what is the rule of thumb? eg 2:1?

"10:1 rule"

It has been suggested that ? degree of ? is lost for every additional covariate added into your model

1 degree of freedom

Assumptions of ANOVA (3)

1. Normality or scores on the DV split by group. 2. Homogeneity of variance (between grps analysis). 3. Sphericity (repeated measures of analysis)

If at the design stage of your research you incorporated a measurement of a variable that you suspected would be related to your DV of interest, you need to investigate if this predicted relationship between the DV and the covariate does in fact exist. How

1. you would do this by plotting the covariate scores against the corresponding DV scores. 2. consider how the scores on this scatterplot correspond to the participants in each of the three independent groups

Number of covariates should be less than (?% of the sample size) minus (number of ? - 1).

10% groups

An ANCOVA addresses the same question as an ANCOVA, however, includes?

ANOVA Q.= Are mean differences between groups on the dependent variable likely to have occurred by chance?' ANCOVA Q.= the same plus the possibility of inter-related variables and/or error playing a significant role in the equation.

For ANCOVA, the question becomes: Are ? differences between groups on the ? dependent variable likely to have occurred by chance?'

Are mean differences between groups on the adjusted dependent variable likely to have occurred by chance?'

The key question addressed by ANOVA is:

Are mean differences between groups on the dependent variable likely to have occurred by chance?'

"What are the consequences of violating the assumption of homogeneity of regression slopes? "When the assumption of homogeneity of regression slopes is met the resulting ?-statistic can be assumed to have the corresponding ?-?; however, when the assumption is not met it can't, meaning that the resulting ?-statistic is being evaluated against a ?.

F, F-distribution, F, distribution different than the one that it actually has

do we want homogeneity or hetrogeneity of regression ?

Homo

Mathematically ANOVA is

Just a special case for multiple regression

do we want the homogeneity of regression slopes assumption to be significant?

NO, if this assumption is not significant homogeneity of regression slopes is not breached and we can run our analysis

What are the two main characteristics of the General Linear Model.

One, that we assume linear relationships between the predictors and the outcome measure. Two we assume that the effects of each predictor are additive with one another.

is the portion of unexplained variance smaller or larger in the ANOVA compared to the ANCOVA

Smaller

If there's only two levels of a categorical predictors what test would we use?

T-test

How do we "trick" the GLM into testing a non-additive (moderation or interaction) effect?

Take the product of two predictors and include that in the interaction term. We multiply the two predictors in our regression model to create an interaction term, and include that to develop a linear equation that can handle non-additive effects for either a moderated multiple regression or a factorial ANOVA. Well done!

You run an ANCOVA analysis, examining the effect of psychological therapy on self-esteem, controlling for self-rated depression in the analysis. You find the following relationship between self-rated depression and self esteem: F(2, 42) = 3.41, p = .04. What does this indicate?

That self-rated depression is related significantly to self-esteem, and makes a significant adjustment to scores of self-esteem

What is the same about an ANOVA to a multiple regression?

The GLM is still linear and additive

The theoretical basis of procedures for t-tests, ANOVA, and ANCOVA is?

The General Linear Model (GLM)

In ANCOVA, when the F statistic for the covariate is significant what does this indicate?

The covariate is significantly related to the DV and will make a significant adjustment to the DV scores.

What is a more flexible and practical approach the the Simple Linear Model? and why?

The general linear model. Because it can explain what other variables might contribute to the model (ie. have more then one variable) and any error.

In an ANOVA model how many explanations/sources of variance are there for the DV and what are they?

Two and Variance explained by the Iv and the portion left unexplained or error variance

How can a GLM accommodate non-linear relationships?

We can transform variables to make them linear and then fit it into the GLM.

In the General Linear Model (Y = B0 + B1X1 + e) Which is the DV, IV, Error and Regression weight

Y= DV, X = IV, e = Error B = Regression weight

Assign variables in the following research question to the GLM: If employee satisfaction is high, is job turnover reduced for both male and female employees? Y? X1? X2? X3?

Y=Job turn over X1=Employee satisfaction X2=Gender X3=Employee satisfaction X Gender

Can the General Linear Model handle non-additive or non-linear effects?

Yes, it can handle it, we just have to trick it, ANOVA will call that an interaction.

Can you have too many covariates?

Yes, it may compromise the statistical power of your analyses

what happens if your f statistic is higher?

a greater likelihood of a significant finding

Overall, ANCOVA is a useful analytical technique that helps explain ? in order to ? the ability to identify an ?

a portion of error variance in the DV . enhance, effect of an IV

Statistically speaking, what is occurring in ANCOVA is that each covariate is added and tested as a predictor of the underlying model. The model is then used to test the difference between DV group means ? for the covariate.

adjusted

What does ANCOVA stand for

analysis of covariance

What does ANOVA stand for

analysis of variance

the smaller the unexplained variance is, the easier it is to?

attain a significant result.

how do you proceed if you have a breach of homogeneity of regression slopes? Basically, you have an interaction present that you j??. You need to acknowledge it and then proceed essentially making your covariate another ? in your analysis. You delve into the interaction that has been obtained, and consider the impact of your original IV of interest at different levels of the covariate.

cannot ignore. IV

What variable do we use in a multiple regression ?

continuous

In an ANOVA, the error is calculated as the difference between the score and the mean of the relative group, the error scores (or residuals) are not at all random. They vary systematically as a function of the ?.

covariate

an ANCOVA is used to determine whether there are any significant differences between two or more groups on a dependent variable, after adjusting for the ?.

covariate

Looking at the output there is only one piece of information you want from this custom model run of your ANCOVA. And that is whether the ?by? interaction term is significant.

covariate by IV

in your ANCOVA, you do want a ? between your covariate and your DV in order to make it worthwhile.

decently strong correlation

at the ? stage of your project, you should strongly consider if to include the ? of a variable or variables that may be related to your ? variable of interest.

design, measurement, dependent

you also dont want ? in covariate scores between your ? groups.

differences, independent

Which form of hypothesis for ANCOVA would be aligned to the use of contrast analysis to appropriately unpack significant main effects?

directional hypothesis

ANCOVA can be used to reduce ? variance and can be used to ? the chance of identifying an effect of the ? variable.

error, enhance, independent

The way you will proceed if you have a breach of homogeneity of regression slopes will differ depending on the nature of your IV and your covariate. But the two options are essentially a ? ?

factorial ANOVA (convert your covariate to a categorical variable and include as an iv) or a moderated regression (keep the covariate continuous and assess the interaction).

ANCOVA is one technique in a family of techniques that in their broadest sense compare ??

group means.

"When a covariate is used we look at its overall relationship with the outcome variable: we ignore the group to which a person belongs. We assume that this relationship between covariate and outcome variable holds true for all groups of participants, which is known as the assumption of

homogeneity of regression slopes.

What will this reduced amount of unexplained variance improve your chance of?

identifying a statistically significant result

Which assumption in checking for ANCOVA's use equates to the sharing of variance between the treatment effect and the covariate?

independence of covariate

Consequently, violating the assumption of homogeneity of regression slopes, the Type I error rate of the test is ? and the power to detect effects is ? (Hollingsworth, 1980). This is especially true when group sizes are ? (Hamilton, 1977) and when the standardized regression slopes differ by more than 0.4 (Wu, 1984)."

inflated, not maximized, unequal

And finally, you do not want a significant ? effect between the ? and the ? on the DV.

interaction, covariate, IV

For an ANCOVA, we need to ensure that the correlation between the DV and covariate is ?,

linear

ANCOVA assumes that the relationship between the DV and the covariate is ?. Secondly, it assumes that the relationship between the DV and the covariate is similar ? ?. This is known as the homogeneity of regression slopes assumption.

linear, across groups

"homogeneity of regression slopes. Think of the assumption like this: imagine a scatterplot for each group of participants with the covariate on one axis, the outcome on the other, and a regression line summarizing their relationship. If the assumption is met then the regression lines should ?

look similar (i.e., the values of b in each group should be equal)."

covariates are not part of the main experimental ?, however they have an influence on the ?

manipulation, DV

ANOVA is used to identify statistically significant differences between the ? of two or more groups

mean

In the GLM, when we trick it to test for non-linear effects or non-additive effects, In multiple regression we're going to call that ?, and factorial ANOVA we're going to tall that ?.

moderation, interaction

"Although in a traditional ANCOVA heterogeneity of regression slopes is a bad thing (Jane Superbrain Box 13.2), there are situations where you might expect regression slopes to differ across groups and that variability may be interesting. "If you have violated the assumption of homogeneity of regression slopes, or if the variability in regression slopes is an interesting hypothesis in itself, then you can explicitly model this variation using?

multilevel linear models "

should our covariate be related to the IV? the covariate shares its variance with?

no only to the DV

In a Simple Linear Model, one variable would?

one variable (e.g. height) would simply explain or predict or otherwise directly relate to another variable (e.g. weight)

t-tests, ? ANOVA/ANCOVA, and ? ANOVA/ANCOVA all share the same analysis goal: determining the significance of mean group differences.

one-way, factorial

Ideally, your covariate should be independent of your IV. Why?

so to specifically reduce the amount of variance that is classified as error.

What does the covariate help explain

some of the previously unexplained, or error in the DV.

Assumption of ANOVA continued You can also run ANCOVA for repeated measures analyses in which case ? may also be of concern.

sphericity

Describe the GLM

the underlying mathematical framework of these tests

When is analysis of variance (ANOVA) most appropriate? And what sort of IV/predictors?

when we true independent variables. Categorical predictors

What is a covariate

A covariate is a variable that we believe is correlated with the dependent variable.

Why would we call an ANOVA a factorial ANOVA?

Because it has multiple predictors

to assess the homogeneity of regression slopes assumption, we can investigat it using a scatterplot of the covariate against the DV in order to see if a linear relationship exists for each group. If the relationship is not linear an ANCOVA is not a suitable analysis. However, if it is relevant to your data set, you may be able to ????

transform the covariate to have a linear relationship with the DV.

violating the assumption of homogeneity of regression slopes, the Type I error rate of the test occurs and the ability to detect an accurate effect occures. this is esspcailly true when?. Group sizez are? and when the standardized regression slopes differ by more than ?(Wu, 1984)."

unequal, 0.4

PRT 2) how can we test whether we have a breach homogeneity after getting the main effects we need to get the interactions. And to test this assumption we need to get an interaction with the covariate. If you navigate your way to the ???? box in SPSS, you'll need to pop your DV in the DV spot, your IV in the ?? spot and your covariate into the covariate spot. Then click on model.

univariate general linear model, fixed factor

What analysis does the GLM cover?

ANOVA, Factorial ANOVA, T-test

Two main reasons to do an ANCOVA

"main reasons to include covariates in ANOVA: To reduce within-group error variance: When we predict an outcome from group means (e.g., when these represent the effect of an experiment), we compute an F-statistic by comparing the amount of variability in the outcome that the experiment can explain against the variability that it cannot explain. If we can attribute some of this 'unexplained' variance (SSR) to other measured variables (covariates), then we reduce the error variance, allowing us to assess more sensitively the difference between group means (SSM). Elimination of confounds: In any experiment, there may be unmeasured variables that confound the results (i.e., variables other than the experimental manipulation that affect the outcome variable). If any variables are known to influence the outcome variable being measured, then including them as covariates can remove these variables as potential explanations for the effect of interest."

You also do not want there to be a relationship between your ? and your ?,

IV, covariate

if your sample size is 200 divided into three groups, use of the 10:1 rule would result in a recommendation that no more than 18 covariates be included in your analyses.

18 = 10% 200 is 20 & number of groups is 3 - 1 is 2 so 2 minus 20 is 18.

What are the two additional ANCOVA-specific data cleaning principles:

7. Independence of covariate and treatment effects, and 8. Homogeneity of regression slopes.

With the use of a GLM, an outcome variable is potentially explained by?

A combination of explanatory variables and error, e.g. y = x1 + x2 + x3 + error.

Is the portion of unexplained variance smaller in the ANOVA or ANCOVA? why

ANCOVA, because of the covariate

Data Cleaning process, Independence of covariate and treatment effects means?

Checking that the covariate and IV are independent

In ANCOVA, the covariates should be:

Correlated with the DV

What is the new variable added into the ANCOVA

Covariant

You will need to create an interaction term, to do this, when you click on model. You'll see, that when you first open this box that a full factorial model will be selected (Top). If you select instead the ??, this will enable you to create your own main effects and interactions. To create an interaction term, you simply highlight the variables you want to combine in an interaction term together and then hit the arrow key to move it across. In our case you would select the IV and the ?. Double check that in the model box you have a term that is your covariate multiplied by your IV. It will have an ? in it to denote that it is a product term. Also move across your IV and covariate ? by ? to create main effects.

Custom model. covariate. asterisk. one one

Why is it important that homogeneity of regression slopes assumption is met accross al groups?

This is important because the ANCOVA analysis bases its calculations on the assumption that it can apply the same regression slope for the DV-covariate relationship to all groups. It can only do this validly if that regression slope is similar between the groups.

In an ACNOVA model how many sources of variance are there for the DV and what are they?

Three and 1. Variance explained by the Iv 2. Variance explained by the covariate. 3. the portion left unexplained or error variance

Using the simple linear model give an example of how height would explain or predict or otherwise directly relate to weight?

a "tall" group would weigh significantly more than a "short" group.

what makes an ANOVA different from multiple regression ? Ie what sort of variable would we use?

a categorical variable

The primary aim of an ANCOVA analysis is to assess relationships between a ? dependent variable, one or more ? independent variables, and one or more continuous covariate variables.

continuous, categorical, continuous

PRT 1) how can we test whether we have a breach homogeneity or of regression assumption? To do this we need to examine the ? between the IV and covariate on the DV. In SPSS, we can do this by running a ? of the ANCOVA. Automatically in SPSS, any variables that you place in the ? ? ?, that is your IVs, SPSS will provide you with a combination of ? ? and ? as a default. This is known as a ???. Variables included in the covariates box are given main effects tests but not

interaction, variation, Fixed Factor box, main effects & interactions, full factorial model

What are the two main things to remember about this framework?

it is linear and additive, but we can trick it to test for non-linear effects or non-additive effects.

covariates are not part of the main experimental manipulation, however they have an influence on the dependent variable (DV). The example; research examining the effects of pharmaceuticals on libido. The primary variables of interest are the IV (?); in this case Viagra) and the DV (?). However, there are many factors outside of pharmaceutical factors that influence libido. Well-designed research would therefore include statistical measures of the influence of these other factors in an attempt to capture a more accurate representation of the relationship between pharmaceuticals and libido. Field cites antidepressants, the contraceptive pill, fatigue, and partner's libido as other potentially important libido influencing factors that should be measured in this research - and then included as ? in the analysis - to more accurately capture the relationship between Viagra and libido.

iv) pharmaceuticals, DV libido, covariates

In ANCOVA, we introduce a ? variable as a ? to help explain some of the variation in the dependent variable.

predictor, covariate

two reasons why an ANCOVA is good to use

reduce error and increase chance of identifying on effect on the DV