Exam 3

Question 1

The overall Fobserved in an Analysis of Regression is F(4,203) = 5.89.
a. How many predictors are there in this regression analysis?

Answer 1

4

Question 2

The overall Fobserved in an Analysis of Regression is F(4,203) = 5.89.
b. How many degrees of freedom are there for the t-test for the significance of each predictor?

Answer 2

206

Question 3

Suppose the observed value of a t-test for a single regression coefficient. If t(39) = 3.00, for alpha=.05, two tailed. If the test of this regression coefficient were reported as an F test instead, what would be the numerical value of Fobserved? What would be the degrees of freedom of this F test?

Answer 3

F (1, 39) = 9.00

Question 4

Suppose you have a categorical variable with four groups, e.g. four regions of the country. You wish to use it as a predictor in a regression analysis. What is the general strategy for employing a categorical variable as a predictor in a regression analysis.

Answer 4

We create g-1 code variables to represent g groups so that each grou phas its own code. We treat them by coding variables that carry the information about the group membership of each case. There are numerous coding schemes that we use. Each coding system carries all the group information and represents the same nominal variable.

Question 5

In any coding scheme for g groups, how many codes are required to characterize the g groups?

Answer 5

G - 1

Question 6

A dummy variable coding scheme with group 3 as the baseline group

Answer 6

Group # C1 C2
1 1 0
2 0 1
3 (base) 0 0

Question 7

an unweighted effects coding scheme with group 3 as the baseline group. interpret coefficients:
Yhat = b1UE1 + b2UE2 + b0

Answer 7

Group #	C1	C2
1		1	0
2		0	1
3 (base)	-1	-1
b1=mean of group 1 minus grand mean
b2=mean of group 2 minus grand mean

Question 8

a series of orthogonal contrast codes. For example, be able to code these contrasts: contrast code that contrasts the mean of the first group with the average of the means of the second and third groups; a contrast code that contrasts the mean of the second group with the mean of the third group.

Answer 8

Group # C1 C2
1 -2 0
2 1 1
3 1 -1

or

Group # C1 C2
1 1 0
2 -.5 .5
3 -.5 .5

Question 9

For dummy coding, be able to take the general regression equation and the codes and explain what each of the coefficients in the equation is measuring

Answer 9

B0 = value of the mean in the baseline group
B1= the difference between the mean of groups in first contrast
B2= the difference between the mean of groups in second contrast

Question 10

What does it mean if two codes from a coding scheme are orthogonal? Under what condition will the following relationship hold:

r2multiple = r2y,c1 + r2y,c2

Answer 10

Orthogonal means that each code accounts for a portion of variance that does not overlap at all with the other codes in the set. The relationship above will only hold for equal sample sizes and orthogonal contrasts

Question 11

Are dummy codes centered?

Answer 11

Dummy codes are not centered

Question 12

Are the pairs of dummy codes in a dummy variable coding scheme orthogonal?

Answer 12

No, the dummy codes are correlated with one another. They share the same base group. They account for overlapping proportions of variance in Y. You can’t compute the correlation of each dummy code with the criterion.

Question 13

What sort of data configuration lends itself to coding with dummy codes?

Answer 13

A configuration in which there is one definite control group or base group.

Question 14

give the set of unweighted effects codes with group 3 as the baseline group

Answer 14

Group # C1 C2
1 1 0
2 0 1
3 (base) -1 -1

Question 15

For unweighted effects coding, be able to take the general regression equation and the codes and explain what each of the coefficients in the equation is measuring if the groups are of equal size

Answer 15

The intercept = the grand mean of all group means

The regression coefficient for each unweighted effects code is the mean of the group coded 1 minus the grand mean.

Question 16

In what way are unweighted effects codes, applied to equal group size data, intimately related to ANOV.

Answer 16

In ANOVA, all the contrasts that go into computing the SStreatment are of each group mean with the grand mean. Each unweighted effect code in regression provides a measure of the difference between a group mean and the grand mean.

Question 17

Are unweighted effects codes centered for equal group size? for unequal group size?

Answer 17

Yes, unweighted effects codes are centered only if we are dealing with equal sample sizes. If we have unequal group sizes, they can be adjusted to weighted effects codes.

Question 18

Are unweighted effects codes orthogonal?

Answer 18

No. Unweighted effects codes are not orthogonal because the sum of the squared validities does not equal r2multiple.

Question 19

Return to thinking about dummy codes. Consider gender as a dummy coded varaible, 1=male, 0=female. Suppose you have the coefficients for the overall regression equation, where X is continuous and D is a dummy code:

Yhat= b1 X + b2 D  + b3 XD + b0 
Yhat= .4X  + .3 D  + .2XD + 1.5  

Explain what each of the four coefficients (including the intercept) measure.

Answer 19

B0: The intercept for females (the intercept for the group coded zero)
B1: the regression of Y on X for females is .4 (the regression of Y on X in the group coded zero)
B2: the difference in intercepts for males minus females is .3 (the difference in the intercepts for the group coded 1 minus for the group coded 0)
B3: the difference in slopes for males minus females is .2 (the difference in the slopes for the group coded 1 minus for the group coded 0)

Question 20

Be able to write the simple regression equation for the group coded zero (female) from the overall equation.
Yhat= .4X + .3 D + .2XD + 1.5

Answer 20

It would be = .4X + 1.5.

Question 21

How would you get a simple regression for the group coded one (males)

Answer 21

The easiest thing would be to reverse the codes, so that 1=female, 0=male, and rerun the regression equation

Question 22

In coding of a two-group variable, we considered the use of unweighted effect codes (+1,-1) versus the codes (+.5, -.5). In the equation

Yhat= b1 X + b2 C + b3 XC + b0 when using the contrast code versus

Yhat= b1 X + b2 UE + b3 X*UE + b0 when using the unweighted effects codes,

explain how the numerical values of coefficients will change when you switch coding systems. Will the significance of the coefficients change when you change between these two coding systems?

Answer 22

In the case of unweighted effects codes (+1, -1), the coefficient b2 and b3 will change. They will only get us halfway to the next code. Since the coefficients represent a one unit change, it will get us to the value of 0.
In the case of the contrast codes (+.5, -.5), the coefficients won’t change because they still represent a one unit change.

Question 23

What is Type III partialing in SAS GLM, in SPSS GLM? Regression or “unique” partialing in SPSS MANOVA?

Answer 23

When we have unequal n’s, there are options for how we can analyze the data. We can use the “unique” partialing (in SPSS MANOVA) or Type III partialing (in SPSS GLM and SAS GLM) where each sums of squares is reported with all other effects partialed out.
SSA with B and AB partialed out
SSB with A and AB partialed out
SSAB with A and B partialed out.

Question 24

Show the regression equation for a two-group experiment with a continuous variable included. In the analysis of covariance, what is the categorical variable called? the continuous control variables?

Answer 24

The categorical variables are called covariates or control variables. The continuous variable is referred to as the variate (coded variable)

Yhat = b1X1 + b2X2 … bpC + b0

C is the categorical variate
X is a covariate, continuous or categorical.

Question 25

Why is there no correlation between the covariate and the variate in the true experiment? With what two things can the covariate be correlated in a quasi-experiment with nonrandom assignment?

Answer 25

There is no correlation between the covariate and the variate in the true experiment because random assignment should control for that. The covariate can be correlated to the DV. In a quasi-experiment, it’s okay for there to be a correlation between characteristics of the subject and assignment to the treatment. A covariate might be more or less correlated with the criterion Y. It comes about because subjects are not randomly assigned.

Question 26

How is power for the test of treatment affected by the covariate in the true experiment (increase power)? How is power in the quasi-experiment affected by the covariate (may increase or decrease power depending on the relationship of the covariate to treatment and criterion)?

Answer 26

Power is increased because the covariate partials out from the criterion a source of variation that is irrelevant to the predictor, which increases the power of the test for an effect. The ANCOVA with a quasi-experiment might increase or decrease power for the test of the effect depending on the relationship of the covariate to treatment and criterion. It’s possible that the covariate partials out error variation in the criterion or it may partial out pre-existing b/w group differences on the criterion that should not be attributed to the treatment.

Question 27

What is a within class regression line in the ANCOVA? What assumption is made about within class regression lines in analysis of covariance? If the assumption is met, is the treatment effect constant over all levels of the covariate?

Answer 27

Within class regression lines are regressions of the criterion on the covariate in each of the conditions. It assumes homogeneity of within class regression meaning that the b1s are equal in the two groups (the slopes are the same). If that assumption is met, the treatment is constant over all levels (no interaction).

Question 28

Show an alternative arrangement of equation
Yhat = b1 X + b2 C + b3 XC + b0
into a simple regression equation that shows the regression of Y on C at different values of X. This is an arrangement that focuses on differences between the means of the two groups on the dependent variable as a function of X.

Answer 28

Yhat = b1 X + b2 C + b3 XC + b0
Yhat = b2 C + b3 XC + b1 X + b0
Yhat = (b2 + b3 X) C + (b1X + b0)
The simple regression coefficient here (b2 + b3 X) gives the value of the difference between the intercept of the group coded 1 minus the group coded 0, at each specific value of X.

Question 29

What does the Johnson-Neyman procedure test?

Answer 29

This procedure provides cutoff values of X (the covariate) beyond which the treatment effect (the difference in elevation of the two regression lines) is significant. It’s the procedure of testing conditional effects of treatment C at particular values of X. It tests whether two lines have significant differences at various points on X.

Question 30

If you use contrast coding for two groups (group 1 = .5, group 2 = -.5), and the contrast code interacts with the continuous variable in the equation, be able to indicate what each of the regression coefficients in the equation measures

Answer 30

B1: the unweighted mean of the slopes of the groups
B2: the difference between the intercepts of the high group versus the low coded group
B3: the difference between the slopes of the high coded group minus the low coded group
B0: the unweighted mean of the two intercepts

Question 31

What is meant by a fixed effects regression model? Under what condition will all inferences be correct in a fixed effects multiple regression, even if we merely sample cases and observe both X and Y?

Answer 31

Fixed effects regression means that the predictors have specified values that are systematically included in the sample. There is no probability distribution for the predictors. We sample X systematically and observe values of Y. All predictors need to be 100% reliable with zero errors and we decide the range of values for the predictors. If we have multivariate normality, meaning each variable is normally distributed AND conditional for each value on one predictor, then all inferences will be correct.

Question 32

What are components of the residual in a regression equation

Answer 32

The residual contains random variation in Y and specification errors of excluding relevant variables or specifying the wrong form of the relationship of predictors to criterion.

Question 33

What are meant by the mean structure and the variance structure of the regression model?

Answer 33

The mean structure refers to the coefficients and predicted scores. The variance structure refers to the error terms, the MS residual, and the standard error.

Question 34

What two aspects of regression analysis are we concerned about when we consider violations of the assumptions of the OLS regression model?

Answer 34

We are concerned about the estimates of coefficients, estimates of standard errors, and MS residuals

Question 35

Explain conditional variances of Y, given X. What is meant by homoscedasticity? The violation of the assumption is heteroscedasticity. What are the effects of heteroscedasticity on the regression analysis?

Answer 35

The conditional variance of Y for each set of fixed values of predictors is an estimate of the error variance. We assume homoscedasticity. Homoscedasticity states that the conditional variances are equal across each combination of one value on each predictor. Heteroscedasticity leaves the estimates of the regression coefficients unbiased; the standard errors of OLS become biased; the direction of bias depends on the relationship of the error variance to a predictor. If the error variance increases as the predictor increases, the bias is negative and significance is over-estimated. If error variance decreases as predictor increases, the bias is positive, and significance is under estimated.

Question 36

Consider the assumption of normally distributed errors? Where does the assumption come into play? What in the regression analysis is affected by violation of the assumption?

Answer 36

Underlies the tests of significance of the multiple correlation and individual regression coefficients, as well as the confidence intervals. The assumption comes into play in inference. Nonnormality of errors does not create bias in the regression coefficients, but may increase the standard errors relative to what these standard errors would be for OLS estimates if data were normally distributed. T and F tests might be biased by nonnormality.

Question 37

Consider the assumption of independent errors across observations. By independent errors I mean that there is no correlation among observations, no clustering. How can errors become correlated? With nonrepeated measures, the measures may be taken on people within groups. With repeated measures, repeated observations on the same individual over time will be correlated with one another.

Answer 37

Errors can become correlated in quasi experiments and repeated measures. With nonrepeated measures, the measures may be taken on people within groups or who are related. With repeated measures, the repeated observations on the same individual over time will be correlated with one another. Correlated errors will show up in mean structure, but won’t be in variance structure because it is correlated with the treatment. We will under-estimate all the error variance and have a positive bias in all our tests.

Question 38

What is the ICC? What does it measure

Answer 38

The ICC is the intraclass correlation. It is an index of how much clustering there is in the data (non-independence). An ICC of 0 means independence. Even a small correlation will give us alpha inflation

Question 39

What are the effects of correlated errors on the OLS regression analysis?
What alternative regression model is appropriate with clustered data?

Answer 39

The effects of correlated errors (autocorrelation) are: the regression coefficients remain unbiased; the regression coefficients may be highly unstable across replications; and MS residual may substantially underestimate the true amount of residual variance in the population. The sample estimates of the standard errors of the regression coefficients may underestimate the corresponding parameters (t-tests for coefficients are positively biased). The fix is to do multilevel modeling

Question 40

What are kernel density estimates? Are they applied to the distribution of a single variable or to pairs of variables? What do they help to illustrate?

Answer 40

The distribution of a variable represented in a histogram that is highlighted with a nonparametric smooth. They are helpful in identifying skew of distributions and outliers.

Question 41

Normal probability plots (p. 4-5, 8-9 of Plots handout). Are these applied to the distribution of a single variable or to pairs of variables? Describe what is on the X axis and Y axis of normal probability plots. If a variable is normally distributed, how will the plot appear? How will an outlier appear?

Answer 41

Normal probability plots detect nonnormality and outliers in the distribution of a single variable, like the residuals. A set of scores is plotted as a fxn of scores that would have been obtained if the variable were normally distributed. The scores are ranked from lowest to highest. Nonnormality may look like a light tail or heavy tail. If the actual scores are normally distributed, the data points of the graph fall on a straight line. Skewed distributions have one heavy and one light tail. Outliers appear as points toward the upper right or lower left.

Question 42

Explain how partial regression leverage plots (also called added variable plots) are constructed . Can more than one variable be partialed at a time? What information do partial regression leverage plots add to plots of residuals against predicted scores.

Answer 42

The partial regression leverage plots allow you to identify the specific predictor that is leading to the difficulties such as model misspecification. In these plots it is also easier to see how a particular case is distorting the regression coefficients. Case 69 example 6

Question 43

What do we mean by the breakdown point of an estimator? What is the breakdown point for OLS estimators?

Answer 43

The breakdown point is the proportion of outlying points in a sample that it takes to change the values of estimates of regression coefficients away from those that would be obtained if no errant points were present. For OLS estimators, the breakdown point is said to be: 1/n

Question 44

What are the three characterizations of errant data points

Answer 44

leverage
distance
influence

Question 45

In what two ways is the term “outlier” used? How is it used in the context of regression diagnostics?

Answer 45

Outliers are extreme points in a distribution. In multiple regression, outliers are conditional as data points whose Y scores are unexpected, given their X scores or position in the predictor space. These are scores that do not follow the regression model.

Question 46

Explain leverage. Is a model required to measure leverage? Does high leverage necessarily mean that a point is affecting the regression outcome?

Answer 46

Leverage is based on predictors. It is the potential for a point to move the regression line. Is the case extreme on the predictors (no model is needed). High leverage does not necessarily mean that the point is affecting the regression outcome.

Question 47

Explain distance. Is a model required to measure distance? Does high distance necessarily mean that a point is affecting the regression outcome?

Answer 47

Distance is based on the residuals. Is the point extreme on Y, given X. A model is needed. High distance does not necessarily mean that a point is affecting the regression outcome. It can also mean distortion of standard errors

Question 48

Explain influence. Is a model required to measure influence? How does influence relate to leverage and distance. Does high influence necessarily mean that the point is affecting the regression outcome?

Answer 48

Influence is the function of both leverage and distance. These are points that distort the regression equation/surface. A model is needed. High influence means that the point is affecting the regression outcome.

Question 49

What is the general strategy for studying the effect of a point on the regression outcome

Answer 49

We would delete the point and rerun the analysis. This is called DFBETAS: a measure of standardized change in regression coefficients when a case is deleted.

Question 50

What measures are on the main diagonal of the hat matrix?

Answer 50

The hat diagonals measure the length from the centroid of the data (leverage)

Question 51

What is the basis of all measures of distance?

Answer 51

The basis of all measures of distance are residuals

Question 52

What is the problem in regression diagnostics with clusters of errant points?

Answer 52

Clustered errant points can mask each other in diagnostic analysis. Clusters can make it difficult to determine which cases to delete. If you have a cluster of influential cases, all working in the same way to change the regression outcome, if you remove one of the cases, the remaining cases in the clusger will continue to exert influence. Therefore removing a case does not show the impact of the case on the analysis.

Question 53

Define effect size in terms of the null hypothesis.

Answer 53

The effect size is the degree to which the treatment changes treated subjects relative to control subjects. It’s the degree to which phenomenon is present in the population or the degree to which the null hypothesis is false. It’s a measure of how far an effect is from the null value (typically zero) in the population.

Question 54

In general, how is effect size measured? What is d as a measure of effect size? How is effect size defined by Cohen for multiple and partial correlations (squared)?

Answer 54

Effect size is a function of the ratio of systematic variance (of predictors or tx manipulation) relative to error or residual variance). It is the proportion of Y variance accounted for by the source in question relative to the proportion of error. Effect sizes are unitless that do not depend on the scale of measurement.

Effect Size = Systematic variance accounted for / Error variance

Cohen’s d is a measure of the differences between the means of two groups (usually experimental and control) relative to the standard error. It is the systematic difference in the numerator by the random variability in the denominator.

Effect sizes for multiple regression is defined in squared terms f2 . The effect size for multiple and partial correlations is done by dividing the squared multiple correlation of a set of predictors by the equation without that set of predictors (if you are predicting a criterion from a set of predictors B).

Question 55

Approximately what percent of the variance is accounted for in Cohen’s (1988) definition of small, moderate, and large effect size for Pearson product moment correlations? What percent of variance is accounted for in Cohen’s (1988) definition of small, moderate, and large effect size squared multiple correlations?

Answer 55

Pearson product moment correlations of .1 (small), .3 (mod), and .5 (large) account for 1%, 9%, and 25% of the variance respectively.
Multiple or correlations (not squared) correspond to .14, .36, and .51 respectively for effect sizes. This translates to an r2 multiple and accounted variance of 2%, 13%, and 26% respectively.

Question 56

Define the power of a statistical test.

Answer 56

The power of a statistical test is the probability that you actually detect a non-zero effect. It’s the probability of rejecting the null hypothesis, given that the null hypothesis is false.

Question 57

What four factors are involved in any instance of hypothesis testing? How can these factors be used to determine the number of subjects required for a research design?

Answer 57

The four factors involved in statistical inference are PANE
P= power of the test (i.e., the probability of rejecting a false null)
A= the level of significance chosen (alpha)
N= the sample size (n)
E= the effect size of the effect

The experimenter can specify in advance the power desired (.8)
The experimenter can specify the alpha level in advance
Estimates of effect size can come from past research or by using the small, mod, and large values set by Cohen.

Question 58

Assume that with 100% reliable predictors (i.e., no measurement error, reliability of each measure = 1.00), the effect size of an effect in a regression analysis is ƒ2 = .18. If the predictors instead each had reliability of .80, would the effect size remain the same or decrease?

Answer 58

The effect size would decrease. This highlights the discrepancy that may exist between a true effect size in the population, and the estimate of the effect size in a sample with predictors measured with error.

Question 59

If a covariate interacts with the categorical variable in ANCOVA, what does this tell us about the slopes of the within class regression lines? What is the difficulty in coming up with an estimate of the treatment effect in ANCOVA if the covariate interacts with the categorical treatment variable?

Answer 59

If we have an interaction between a continuous and a two-group categorical variable, this means that the regression of Y on X is different within the two groups (that the within class regression slopes differ in the two groups). Violates homogeneity of within class regression and the treatment effect estimate depends on the value of the covariate. It becomes a much more nuanced estimate

Question 60

What does the phrase “measured without error” mean with respect to the reliability of a variable?

Answer 60

Variable is 100% reliable

Question 61

What are meant by attenuation and negative bias with regard to estimates of population parameters.

Answer 61

Attenuation means that the parameter estimate is closer to zero in the sample than in the population (shrinks toward zero). In contrast, negative bias means that the parameter estimate is closer to minus infinity.

Question 62

What is the effect of measurement error in predictors in the one-predictor regression equation?

Answer 62

Measurement error attenuates regression coefficients in the one predictor case

Question 63

What is the effect of measurement error in the multiple regression equation? Is the direction of bias in coefficients known—can it be either positive or negative bias?

Answer 63

Say we have a regression equation: Yhat = b1 X + b2 Z + b0. Measurement error in predictor Z will affect the b2 regression coefficient. The measurement error in Z will also affect the b1 regression coefficient. The direction of bias in the coefficients can be in either direction, that is, to increase the regression coefficient or to decrease the regression constant. In general, the direction of the bias is not known

Question 64

What is the effect of measurement error for the XZ interaction in a regression equation containing X, Z, and XZ as predictors? How is the reliability of the interaction term related to the reliabilities of the individual predictors?

Answer 64

The reliability of XZ interaction term is exactly equal to the product of the reliabilities of the individual variables for uncorrelated X and Z, that is Roe XZ, XZ = Roe XX Roe ZZ. Thus the reliability of the product term is lower than the reliability of the variables of which it is compromised, and the sample size requirement for detecting interactions is large

Question 65

What is the problem with simply dividing a residual by its own standard error to compute a studentized (or internally studentized) residual?

Answer 65

the case can move the regression plane toward itself, thereby reducing its own residual and increasing the residuals for all other cases, yielding larger MS residual.

Question 66

What is the solution to the problem described in Q. 61? How is MSresidual(i) computed? I will refer to a residual divided by its deleted standard error as a studentized deleted or (externally studentized) residual.

Answer 66

Carry out the regression analysis with the cases removed; compute the predicted score and the standard error from the analysis with the case removed

Question 67

What does DFFITS measure? Is there one or more than one measure of DFFITS for each case in a a regression equation?

Answer 67

One measure for the case. The change in the predicted score of a case (in standardized z form) due to the inclusion of the case – shows that the case is moving the regression plane. DFFITS is a global measure of standardized change based on a predicted score.

Question 68

What does DFBETAS measure? Is there one or more than one measure of DFFITS for each case in a a regression equation?

Answer 68

For each case, more than one measure, specifically one measure of how the case is changing each regression equation, including the intercept. The change in each regression coefficient due to the inclusion of the case – shows that the case is changing the regression model. DFBETAS measures the standardized change in regression coefficients when a case is deleted.
Case IN MINUS case OUT

Question 69

What is meant by under-adjustment bias in the analysis of covariance. What produces under-adjustment biase?

Answer 69

Under adjustment bias in the analysis of covariance refers to the failure of the covariate(s) to adjust completely for differences between groups. The under adjustment bias is due to unreliability of the covariates

Question 70

How does measurement error affect the sample size requirements for testing interactions in multiple regression.

Answer 70

Having measurement error in the predictors X and Z entering the XZ interaction greatly increases the sample size needed to dtect a true interaction.