Statistics Final Study #2:

Front 1

In the pretest-posttest with control group experimental design, what does it mean if the interaction is significant?

Back 1

It means that the change in the two groups from pre to post was different - this is what we are interested in.
On a graph, it would look like the control group was flat (no change) and the treatment group depression index dropped from pre to post.

Front 2

What does analysis of covariance do?

Back 2

It reduces the error in order to make the difference between group means a significant difference. IOW: An ANCOVA can give us a more powerful test of the hypothesis that all group means are equal. This is because even when group means appear to be different, they may not be significantly different because error (within-group variability) is large.

Front 3

How does an ANCOVA work?

Back 3

It uses information provided by a continuous variable, called a covariate, to remove some of the variability due to individual differences from the error term.

Front 4

What is a covariate?

Back 4

The covariate is a variable that is correlated with the DV. It can be useful when the magnitude of its correlation with the DV is .3 or greater.

Front 5

ANCOVA example:
Compare two methods of teaching algebra to 8th graders.
IV: method
DV: final exam score
Data for 10 participants:
Scores vary within methods (within group variability)
Mean scores for two methods: 50, 65
ANCOVA allows us to reduce this error and make the F ratio larger.

Back 5

In ANCOVA, each score on the DV (Y), is adjusted for the covariate (X), and then an ANCOVA is run on the adjusted Y scores. Within-group variability for the adjusted Y scores is smaller than the within-group variability of the original Y scores when the covariate is correlated with the DV.

Front 6

In ANOVA:
Y = grand mean + group effect + error
In ANCOVA: ?

Back 6

Y = grand mean + group effect + effect of covariate + error
In ANCOVA, variability due to the covariate is removed from the error term and attributed to the covariate.

Front 7

In ANCOVA, the Y score is adjusted as follows:

Back 7

Adjusted Y = Y - b (X - Xbar)
b is the regression coefficient in the regression of Y on X

Front 8

What does the ANCOVA procedure accomplish?

Back 8

It changes the Y scores to what they would be if the participants were more similar on the covariate (X).
Ex. in a math aptitude test, final exam scores (Y scores) would be adjusted so that they would represent the scores the participants would have received if they had been more similar in math aptitude (covariate) - making them less variable than the actual scores listed.

Front 9

How to use ANCOVA:

Back 9

1. Once we know the regression coefficient, e.g. b = .5
2. Compute the mean scores, e.g., X = 100
3. Compute the adjusted Y score for each individual using the formula: Y = Y - b (X - Xbar)
i.e., plug in X and Y scores
Ex.
(X = 60, Y = 30)
Adj Y = 30 - .5(60 - 100) = 50
(X = 140, Y = 70)
Adj Y = 70 - .5(140 - 100) = 50
Higher scores get reduced, lower scores get increased = variability is reduced.

Front 10

In regression terms,
1. In ANOVA, a specific score Y is predicted from:
2. In ANCOVA, a specific score Y is predicted from:

Back 10

1. The grand mean and group only
2. The grand mean, group and covariate

Front 11

Assumptions of ANCOVA (5)

Back 11

1. All assumptions for regular ANOVA
2. Covariate should be normally distributed
3. The regression of the DV (Y) on the covariate (X) is linear in each group. Check by examining a scatterplot.
4. Homogeneity of regression coefficients. In the regression of Y on X for each group separately, the coefficients (i.e., Beta's for the populations, "b's" for the sample) are all equal. This means that on a graph all groups regression lines are parallel, just starting at different heights on the Y axis
5. The groups do not differ on the covariate, i.e., group means are equal (or not too unequal)

Front 12

Another ANCOVA example:
Comparing three treatments (diet, exercise and medication) to see which is most effective in reducing blood pressure.

Back 12

Because we know blood pressure is related to age, we can include age as a covariate. So, group IV is treatment group; DV is blood pressure; covariate is age

Front 13

Other ANCOVA facts:
1. Contrasts and multiple comparisons can also be run on?

Back 13

1. ANCOVA adjusted scores

Front 14

Other ANCOVA facts:
2. ANCOVA can also be used with _____ designs, and there can be more than one _____.

Back 14

2. Factorial designs, one Covariate

Front 15

What is the predictor variable?

Back 15

The variable from which the prediction is made, e.g., verbal ability

Front 16

What is the outcome variable?

Back 16

The variable which is predicted, e.g., improvement in therapy

Front 17

In computing the correlation coefficient you see the strength and direction of the relationship between the two variables. What are we doing with linear regression?

Back 17

1. Fitting a straight line to the data.
2. Predict from a particular value of verbal ability (ex.) to a particular value of improvement in therapy.

Front 18

Once we have the regression line, what do we know?

Back 18

For an increase of 1 point in X, you will know how much of an increase/decrease in Y.

Front 19

Equation of a straight line:

Back 19

Y = a + bX
a: the value of Y when X = 0
Slope = b = Y2 - Y1/X2 - X1

Front 20

What does the slope tell us?

Back 20

The change is Y relative to X. More specifically, b = the amount that Y changes when X changes by 1 (ex. 2 = 2/1)

Front 21

The technique of linear regression is used to find what?

Back 21

The equation of the line that best fits the data.

Front 22

What does the method of least squares find and how does it do it?

Back 22

Using linear regression, we find a and b by using the data collected on the variables X and Y for a number of participants.
We use the method of least squares to find the line.
This method minimizes the average squared distance from the points to the regression line.
The distance between each point (Y) and the line (Y-hat) is squared; then the squares are summed.

Front 23

What is the regression equation for Y-hat?

Back 23

Y-hat = a + bX

Front 24

What are the two ways to find the predicted value of Y?

Back 24

1. Substitute a particular value of X into the regression equation and solve for Y
2. Draw the regression line on a graph and read the predicted value of Y from the graph

Front 25

With standard scores, the slope equals what?

Back 25

The correlation coefficient.

Front 26

Linear regression is appropriate only when the relationship between X and Y is:

Back 26

Linear

Front 27

The higher the correlation between X and Y:

Back 27

The better the regression line will fit the data. The correlation is an index of fit.

Front 28

It is also possible to predict X from Y. The resulting regression equation is slightly different because why?

Back 28

The sums of squares are minimized horizontally rather than vertically.

Front 29

What is the difference between linear regression and multiple regression?

Back 29

More than one predictor may be used to predict Y.

Front 30

Another name for predictors is ____ and there is a relationship between several predictors and one single ____ of interest.

Back 30

Factors; Criterion

Front 31

Multiple regression is just a simple and accurate way to:

Back 31

Combine information about several factors (predictors) to make the best possible predictions.

Front 32

What is the multiple regression equation?

Back 32

Y-hat = a + (bsub1)(Xsub1) + (bsub2)(Xsub2) + (bsub3)(Xsub3) +...

Front 33

Example equation:
Y-hat = -3 + 5(Xsub1) + 4(Xsub2)

Back 33

To get a predicted Y score for Carol, we would subsititute her scores:
Y-hat = -3 + 5(0) + 4(2)
=-3 + 0 + 8 = 5

Front 34

Notation
1. Y-hat

Back 34

1. Y-hat: the predicted Y score

Front 35

Notation
2. R-squared

Back 35

2. R-squared: The multiple coefficient of determination

Front 36

Notation:
3. R

Back 36

3. R: + square root(R-squared) = The multiple correlation coefficient or coefficient of multiple correlation

Front 37

Notation:
4. Y - Y-hat

Back 37

4. Y - Y-hat: The residual, i.e., how far off the predicted Y is from the actual Y

Front 38

T or F: Regression coefficients are generally not comparable

Back 38

True. Only when all the predictors all have the same SD is this possible.

Front 39

What is another name for Standardized Regression Coefficients?

Back 39

Beta Weights

Front 40

What is the purpose of Standardized Regression Coefficients?

Back 40

To compare the strength of predictors - Beta Weights can be compared to find out the relative strengths of the different predictors.

Front 41

How do Standardized Regression Coefficients allow you to compare the strength of predictors?

Back 41

Standardizing each of the variables guarantees that all variables have the same SD of 1, and that the intercept in the regression equation is 0.
Indicated by "STD"

Front 42

Example
Al's regression equation:
STD Y-hat = 0 + .218(STD Xsub1) + .436(STD Xsub2) + .873(STD Xsub3)

Back 42

To estimate Al's standard score on Y, you add .218 times Al's standard score on Xsub1 and .436 times his standard score on Xsub2 and .873 times his standard score on Xsub3.
IN this example, we can compare the .436 beta coefficient for Xsub2 with the .218 beta coefficient for Xsub1 and conclude that "in the multiple regression using X1, X2 and X3 to predict Y, the contribution of the variable X2 is twice that of the variable X1."

Front 43

What is the best estimate of population R-squared?

Back 43

Adjusted R-squared based on sample data.

Front 44

R-squared-adjusted equation:

Back 44

See Multiple Regression handout p.5

Front 45

The multiple regression technique is a powerful way to do what?

Back 45

Predict scores on a dependent variable.

Front 46

In a small sample, if you use several independent variables, you will automatically obtain a high multiple correlation in the sample, even if what?

Back 46

There is no relationship in the population.

Front 47

What does R-squared-adjusted do?

Back 47

It adjusts for the "ad hoc" aspect of multiple correlation, reducing the estimate, which makes it more conservative and more realistic.

Front 48

How can you guarantee that your sample R-squared is fairly close to R-squared-adjusted?

Back 48

Have at least 10 cases in your sample for each independent variable. i.e., if you make "n" equal to at least 10 times "k", the sample R-squared will almost never be unrealistically high - no "over-fitting"

Front 49

How do you use an ANOVA to test hypotheses about R?

Back 49

After computing sample R, you want to ask, "Is this convincing evidence that the population R is greater than zero?" Or, "Is this convincing evidence that in the population, X1 and X2 are related to Y?"
Null Hyp: Population R = 0
Alt Hyp: Population R not = 0
Obtain the F-ratio for the regression and compare it to a critical value to determine significance.

Front 50

What is "Part Correlation" or "Semipartial Correlation"?

Back 50

When we are only interested in how much additional explanation we will get if we add another independent variable.

Front 51

Part Correlation Example:
To assess the incremental validity of a new scale, we ask how much additional predictive power will we gain if we add the scale to an existing test battery.
If we first tried to predict Y using X only, what proportion of the variation in Y could we explain using X alone?
R-squared = (-.160)squared = .026; this leaves about .974 of the variation unexplained.

Back 51

The variable Z can explain all of the remaining variation - Z can explain the remaining .974 of the variation in Y that couldn't be explained by X. So it makes sense that the "Part Correlation" for Z is the square root of .974, or about +-.987

Front 52

What is Simultaneous (Standard) Multiple Regression

Back 52

The researcher specifies one set of predictors (IV's) and they are all used to produce the single best multiple regression equation for predicting scores on the criterion (DV).

Front 53

What is a Hierarchical/Sequential Multiple Regression?

Back 53

Two or more multiple regressions are run, with an increasing number of IV's.

Front 54

How is a Hierarchical/Sequential Multiple Regression run?

Back 54

The order in which IV's are added is determined in advance by the researcher.
A common strategy is to first specify that "nuisance" variables be entered first. Then the researcher can add the actual IV's of interest to see if an increase in prediction (a change in R-squared) is obtained.

Front 55

Why is a Hierarchical/Sequential Multiple Regression a good way to statistically control for the effect of nuisance variables?

Back 55

If the IV's produce a strong increase in prediction, after the nuisance variables have already been included in the regression equation, we can be sure that the relationship that is discovered is not really due to the effect of the nuisance variables.

Front 56

Example: Hierarchical/Sequential Multiple Regression

Back 56

If we wanted to determine whether stress level and coping will predict illness, once age and SES have been controlled, we would:
1. Run a multiple regression to predict illness from age and SES
2. Then add stress level and coping, and see of the prediction is significantly improved

Front 57

What is Stepwise/Statistical Regression?

Back 57

Two or more multiple regression are run, with an increasing number of IV's.

Front 58

How is a Stepwise/Statistical Regression run?

Back 58

The order in which the IV's are added is determined by statistical criteria.
On each step, the variable is added to the regression equation that contributes most (and significantly) to improving prediction, i.e., produces the largest increase in R-squared.
A variable can also be deleted if it no longer significantly contributes to prediction.

Front 59

In multiple regression, what types of IV's can there be?

Back 59

Interval/ratio or Nominal (e.g., religion - must be Dummy coded)

Front 60

What is Multicollinearity (Collinearity)

Back 60

If one or more IV's could have been predicted very well using the other IV's, multicollinearity is said to exist, i.e., redundancy or highly correlated predictor variables.

Front 61

Why is Multicollinearity problematic?

Back 61

It causes problems in the interpretation of regression coefficients and may make it possible to carry out the regression.