Chapter8

Question 1

Adjusted predicted value

Answer 1

a measure of the influence of a particular case of data. It is the predicted value of a case from a model estimated without that case included in the data. The value is calculated by re-estimating the model without the case in question, then using this new model to predict the value of the excluded case. If a case does not exert a large influence over the model then its predicted value should be similar regardless of whether the model was estimated including or excluding that case. The difference between the predicted value of a case from the model when that case was included and the predicted value from the model when it was excluded is the DFFit.

Question 2

Adjusted R²

Answer 2

a measure of the loss of predictive power or shrinkage in regression. The adjusted R² tells us how much variance in the outcome would be accounted for if the model had been derived from the population from which the sample was taken.

Question 3

Autocorrelation

Answer 3

when the residuals of two observations in a regression model are correlated.

Question 4

bi

Answer 4

unstandardized regression coefficient. Indicates the strength of relationship between a given predictor, i, of many and an outcome in the units of measurement of the predictor. It is the change in the outcome associated with a unit change in the predictor.

Question 5

βi

Answer 5

standardized regression coefficient. Indicates the strength of relationship between a given predictor, i, of many and an outcome in a standardized form. It is the change in the outcome (in standard deviations) associated with a one standard deviation change in the predictor.

Question 6

Cook’s distance

Answer 6

a measure of the overall influence of a case on a model. Cook and Weisberg (1982) have suggested that values greater than 1 may be cause for concern.

Question 7

Covariance ratio (CVR)

Answer 7

a measure of whether a case influences the variance of the parameters in a regression model. When this ratio is close to 1 the case has very little influence on the variances of the model parameters. Belsey et al. (1980) recommend the following: if the CVR of a case is greater than 1 + [3(k + 1)/n] then deleting that case will damage the precision of some of the model’s parameters, but if it is less than 1 ‚àí [3(k + 1)/n] then deleting the case will improve the precision of some of the model’s parameters (k is the number of predictors and n is the sample size).

Question 8

Cross-validation

Answer 8

assessing the accuracy of a model across different samples. This is an important step in generalization. In a regression model there are two main methods of cross-validation: adjusted R² or data splitting, in which the data are split randomly into two halves, and a regression model is estimated for each half and then compared.

Question 9

Deleted residual

Answer 9

a measure of the influence of a particular case of data. It is the difference between the adjusted predicted value for a case and the original observed value for that case.

Question 10

DFBeta

Answer 10

a measure of the influence of a case on the values of bi in a regression model. If we estimated a regression parameter bi and then deleted a particular case and re-estimated the same regression parameter bi, then the difference between these two estimates would be the DFBeta for the case that was deleted. By looking at the values of the DFBetas, it is possible to identify cases that have a large influence on the parameters of the regression model; however, the size of DFBeta will depend on the units of measurement of the regression parameter.

Question 11

DFFit

Answer 11

a measure of the influence of a case. It is the difference between the adjusted predicted value and the original predicted value of a particular case. If a case is not influential then its DFFit should be zero - hence, we expect non-influential cases to have small DFFit values. However, we have the problem that this statistic depends on the units of measurement of the outcome and so a DFFit of 0.5 will be very small if the outcome ranges from 1 to 100, but very large if the outcome varies from 0 to 1.

Question 12

Dummy variables

Answer 12

a way of recoding a categorical variable with more than two categories into a series of variables all of which are dichotomous and can take on values of only 0 or 1. There are seven basic steps to create such variables: (1) count the number of groups you want to recode and subtract 1; (2) create as many new variables as the value you calculated in step 1 (these are your dummy variables); (3) choose one of your groups as a baseline (i.e., a group against which all other groups should be compared, such as a control group); (4) assign that baseline group values of 0 for all of your dummy variables; (5) for your first dummy variable, assign the value 1 to the first group that you want to compare against the baseline group (assign all other groups 0 for this variable); (6) for the second dummy variable assign the value 1 to the second group that you want to compare against the baseline group (assign all other groups 0 for this variable); (7) repeat this process until you run out of dummy variables.

Question 13

Durbin-Watson test

Answer 13

a test for serial correlations between errors in regression models. Specifically, it tests whether adjacent residuals are correlated, which is useful in assessing the assumption of independent errors. The test statistic can vary between 0 and 4, with a value of 2 meaning that the residuals are uncorrelated. A value greater than 2 indicates a negative correlation between adjacent residuals, whereas a value below 2 indicates a positive correlation. The size of the Durbin-Watson statistic depends upon the number of predictors in the model and the number of observations. For accuracy, look up the exact acceptable values in Durbin and Watson’s (1951) original paper. As a very conservative rule of thumb, values less than 1 or greater than 3 are definitely cause for concern; however, values closer to 2 may still be problematic depending on the sample and model.

Question 14

F-ratio

Answer 14

a test statistic with a known probability distribution (the F-distribution). It is the ratio of the average variability in the data that a given model can explain to the average variability unexplained by that same model. It is used to test the overall fit of the model in simple regression and multiple regression, and to test for overall differences between group means in experiments.

Question 15

Generalization

Answer 15

the ability of a statistical model to say something beyond the set of observations that spawned it. If a model generalizes it is assumed that predictions from that model can be applied not just to the sample on which it is based, but to a wider population from which the sample came.

Question 16

Goodness of fit

Answer 16

an index of how well a model fits the data from which it was generated. It’s usually based on how well the data predicted by the model correspond to the data that were actually collected.

Question 17

Hat values

Answer 17

another name for leverage.

Question 18

Heteroscedasticity

Answer 18

the opposite of homoscedasticity. This occurs when the residuals at each level of the predictor variables(s) have unequal variances. Put another way, at each point along any predictor variable, the spread of residuals is different.

Question 19

Hierarchical regression

Answer 19

a method of multiple regression in which the order in which predictors are entered into the regression model is determined by the researcher based on previous research: variables already known to be predictors are entered first, new variables are entered subsequently.

Question 20

Homoscedasticity

Answer 20

an assumption in regression analysis that the residuals at each level of the predictor variable(s) have similar variances. Put another way, at each point along any predictor variable, the spread of residuals should be fairly constant.

Question 21

Independent errors

Answer 21

for any two observations in regression the residuals should be uncorrelated (or independent).

Question 22

Leverage

Answer 22

leverage statistics (or hat values) gauge the influence of the observed value of the outcome variable over the predicted values. The average leverage value is (k+1)/n in which k is the number of predictors in the model and n is the number of participants. Leverage values can lie between 0 (the case has no influence whatsoever) and 1 (the case has complete influence over prediction). If no cases exert undue influence over the model then we would expect all of the leverage values to be close to the average value. Hoaglin and Welsch (1978) recommend investigating cases with values greater than twice the average (2(k + 1)/n) and Stevens (2002) recommends using three times the average (3(k + 1)/n) as a cut-off point for identifying cases having undue influence.

Question 23

Mahalanobis distances

Answer 23

these measure the influence of a case by examining the distance of cases from the mean(s) of the predictor variable(s). One needs to look for the cases with the highest values. It is not easy to establish a cut-off point at which to worry, although Barnett and Lewis (1978) have produced a table of critical values dependent on the number of predictors and the sample size. From their work it is clear that even with large samples (N = 500) and five predictors, values above 25 are cause for concern. In smaller samples (N = 100) and with fewer predictors (namely three) values greater than 15 are problematic, and in very small samples (N = 30) with only two predictors values greater than 11 should be examined. However, for more specific advice, refer to Barnett and Lewis’s (1978) table.

Question 24

Mean squares

Answer 24

a measure of average variability. For every sum of squares (which measure the total variability) it is possible to create mean squares by dividing by the number of things used to calculate the sum of squares (or some function of it).

Question 25

Model sum of squares

Answer 25

a measure of the total amount of variability for which a model can account. It is the difference between the total sum of squares and the residual sum of squares.

Question 26

Multicollinearity

Answer 26

a situation in which two or more variables are very closely linearly related.

Question 27

Multiple R

Answer 27

the multiple correlation coefficient. It is the correlation between the observed values of an outcome and the values of the outcome predicted by a multiple regression model.

Question 28

Multiple regression

Answer 28

an extension of simple regression in which an outcome is predicted by a linear combination of two or more predictor variables. The form of the model is: (see above image)

Question 29

Ordinary least squares (OLS)

Answer 29

a method of regression in which the parameters of the model are estimated using the method of least squares.

Question 30

Outcome variable

Answer 30

a variable whose values we are trying to predict from one or more predictor variables.

Question 31

Perfect collinearity

Answer 31

exists when at least one predictor in a regression model is a perfect linear combination of the others (the simplest example being two predictors that are perfectly correlated - they have a correlation coefficient of 1).

Question 32

Predicted value

Answer 32

the value of an outcome variable based on specific values of the predictor variable or variables being placed into a statistical model.

Question 33

Predictor variable

Answer 33

a variable that is used to try to predict values of another variable known as an outcome variable.

Question 34

Residual

Answer 34

The difference between the value a model predicts and the value observed in the data on which the model is based. Basically, an error. When the residual is calculated for each observation in a data set the resulting collection is referred to as the residuals.

Question 35

Residual sum of squares

Answer 35

a measure of the variability that cannot be explained by the model fitted to the data. It is the total squared deviance between the observations, and the value of those observations predicted by whatever model is fitted to the data.

Question 36

Shrinkage

Answer 36

the loss of predictive power of a regression model if the model had been derived from the population from which the sample was taken, rather than the sample itself.

Question 37

Simple regression

Answer 37

a linear model in which one variable or outcome is predicted from a single predictor variable. The model takes the form: (see above image)

Question 38

Standardized DFBeta

Answer 38

a standardized version of DFBeta. These standardized values are easier to use than DFBeta because universal cut-off points can be applied. Stevens (2002) suggests looking at cases with absolute values greater than 2.

Question 39

Standardized DFFit

Answer 39

a standardized version of DFFit.

Question 40

Standardized residuals

Answer 40

the residuals of a model expressed in standard deviation units. Standardized residuals with an absolute value greater than 3.29 (actually, we usually just use 3) are cause for concern because in an average sample a value this high is unlikely to happen by chance; if more than 1% of our observations have standardized residuals with an absolute value greater than 2.58 (we usually just say 2.5) there is evidence that the level of error within our model is unacceptable (the model is a fairly poor fit of the sample data); and if more than 5% of observations have standardized residuals with an absolute value greater than 1.96 (or 2 for convenience) then there is also evidence that the model is a poor representation of the actual data.

Question 41

Stepwise regression

Answer 41

a method of multiple regression in which variables are entered into the model based on a statistical criterion (the semi-partial correlation with the outcome variable). Once a new variable is entered into the model, all variables in the model are assessed to see whether they should be removed.

Question 42

Studentized deleted residual

Answer 42

a measure of the influence of a particular case of data. This is a standardized version of the deleted residual.

Question 43

Studentized residuals

Answer 43

a variation on standardized residuals. A Studentized residual is an unstandardized residual divided by an estimate of its standard deviation that varies point by point. These residuals have the same properties as the standardized residuals but usually provide a more precise estimate of the error variance of a specific case.

Question 44

Suppressor effects

Answer 44

situation where a predictor has a significant effect, but only when another variable is held constant.

Question 45

t-statistic

Answer 45

Student’s t is a test statistic with a known probability distribution (the t-distribution). In the context of regression it is used to test whether a regression coefficient b is significantly different from zero; in the context of experimental work it is used to test whether the differences between two means are significantly different from zero. See also paired-samples t-test and Independent t-test.

Question 46

Tolerance

Answer 46

tolerance statistics measure multicollinearity and are simply the reciprocal of the variance inflation factor (1/VIF). Values below 0.1 indicate serious problems, although Menard (1995) suggests that values below 0.2 are worthy of concern.

Question 47

Total sum of squares

Answer 47

a measure of the total variability within a set of observations. It is the total squared deviance between each observation and the overall mean of all observations.

Question 48

Unstandardized residuals

Answer 48

the residuals of a model expressed in the units in which the original outcome variable was measured.

Question 49

Variance inflation factor (VIF)

Answer 49

a measure of multicollinearity. The VIF indicates whether a predictor has a strong linear relationship with the other predictor(s). Myers (1990) suggests that a value of 10 is a good value at which to worry. Bowerman and O’Connell (1990) suggest that if the average VIF is greater than 1, then multicollinearity may be biasing the regression model.