ANOVA 1

Objectives

What is the difference between a continuous variable and a categorical variable? Give a concrete example of each.

What is dummy [effect, orthogonal] coding?

What do the intercept and b weights mean for these models?

Why might I choose one model rather than another (that is, choose either dummy, effect or orthogonal coding) to analyze my data?

Is there any advantage to using regression rather than some ANOVA program to analyze designs with categorical IVs? What effect does unbalanced (unequal) cell size have on the interpretation of dummy [effect, orthogonal] coded regression slopes and intercepts?

 Dummy Coding Regression Formulas Effect Coding Orthogonal coding Unequal Sample Sizes

Categorical IVs: Dummy, Effect, & Orthogonal Coding

What we are doing here is ANOVA with regression techniques; that is, we are analyzing categorical (nominal) variables rather than continuous variables. There are some advantages to doing this, especially if you have unequal cell sizes. The computer will be doing the work for you. However, I want to show you what happens with the 3 kinds of coding so you will understand it. You are already familiar with ANOVA, so I am not going to discuss it. We are going to cover lots of ground quickly here. This is designed merely to familiarize you with the correspondence between regression and analysis of variance. Both methods are specific cases of a larger family called the general linear model.

Dummy Coding

With this kind of coding, we put a '1' to indicate that a person is a member of a category, and a '0' otherwise. Category membership is indicated in one or more columns of zeros and ones. For example, we could code sex as 1=female 0=male or 1=male 0=female. If we did, we would have a column variable indicating status as male or female. Or we could code for marital status as 1=single 0=married or 1=married 0=single. Ordinarily if we wanted to test for group differences, we would use a t-test or an F-test. But we can do the same thing with regression. Let's suppose we want to know whether people in general are happier if they are married or single. So we take a small sample of people shopping at University Square Mall and promise them some ice cream if they fill out our life satisfaction survey, which some do. They also fill out some demographic information, an item of which is marital status (Status), which we code 1=single 0=married. For fun, let's see what happens if we code it the other way (Status 2; 0=single 1=married) Our data:

 Status Satisfaction Status2 Single 1 25 0 S 1 28 0 S 1 20 0 S 1 26 0 S 1 25 0 M = 24.8 SD = 2.95 N=5 Married 0 30 1 M 0 28 1 M 0 32 1 M 0 33 1 M 0 28 1 M = 30.20 SD = 2.28 N=5 M .5 27.5 .5 SD .53 3.78 .53

 Sat Grand mean Dev Dev2 Cell Mean Dev Dev2 25 27.5 -2.5 6.25 24.8 0.2 .04 28 27.5 0.5 0.25 24.8 3.2 10.24 20 27.5 -7.5 56.25 24.8 -4.8 23.04 26 27.5 -1.5 2.25 24.8 1.2 1.44 25 27.5 -2.5 6.25 24.8 0.2 .04 30 27.5 2.5 6.25 30.2 -.2 .04 28 27.5 0.5 0.25 30.2 -2.2 4.48 32 27.5 4.5 20.25 30.2 1.8 3.24 33 27.5 5.5 30.25 30.2 2.8 7.84 28 27.5 0.5 0.25 30.2 -2.2 4.84 Sum 275 0 128.5 275 0 55.60

We have 10 people, 5 each in two groups. The sum of squared deviations from the grand mean is 128.5 (SStot); the sum of squared deviations from the cell means is 55.60 (SSwithin), and difference must be SSbetween = 128.5-55.60 = 72.90. To test for the difference we find the ratio of the two mean squares:

Or we could compute a t-test by

And if we square this result, we get 10.49, which is our value for F (recall that F = t2).

To compute regressions, we find that:

 X Mean X x x2 Y Mean Y y xy 1 0.5 0.5 0.25 25 27.5 -2.5 -1.25 1 0.5 0.5 0.25 28 27.5 0.5 0.25 1 0.5 0.5 0.25 20 27.5 -7.5 -3.75 1 0.5 0.5 0.25 26 27.5 -1.5 -0.75 1 0.5 0.5 0.25 25 27.5 -2.5 -1.25 0 0.5 -0.5 0.25 30 27.5 2.5 -1.25 0 0.5 -0.5 0.25 28 27.5 0.5 -0.25 0 0.5 -0.5 0.25 32 27.5 4.5 -2.25 0 0.5 -0.5 0.25 33 27.5 5.5 -2.75 0 0.5 -0.5 0.25 28 27.5 0.5 -0.25 Sums 5 5 0 2.5 275 275 0 -13.5

Regression Formulas

 Formula Status Status2 -13.5/2.5 = -5.4 13.5/2.5 = 5.4 27.5-(-5.4*.5)= 30.20 27.5-(5.4*.5) = 24.8 Y' =30.20-5.4X Y'=24.8+5.4X -5.4(-13.5)=72.90 5.4(13.5)=72.90 128.5-72.90=55.6 128.5-72.9=55.6 55.6/8 = 6.95 55.6/8=6.95 Sqrt(6.95/2.5) = 1.667 Sqrt(6.95/2.5) = 1.667 -5.4/1.667 = -3.239 5.4/1.667 = 3.239 72.9/128.5=.5673 72.9/128.5=.5673 (.57/1)/(.43/8)= 10.49 (.57/1)/(.43/8)= 10.49

Points to notice:

1. If we switch the vector (column) from zeros to ones (status to status 2), we just change the sign of the deviations from the mean. This gives us b weights of equal magnitude but opposite signs for the two analyses (status and status 2). The bulk of the regression results are identical for the two methods of coding the category. This means that we get essentially the same results regardless of which group gets 1s and which gets 0s.
2. The regression equations are fit to data for zeros and ones -- the X variable only takes on these two values.
3. When X = 0, our predicted value is the mean for that group (those designated with a zero). When X = 1, our predicted value is the mean for that group. Look at the regression equations for each. When single was coded 1 (status), the equation was Y' = 30.2 - 5.4X. So when X is zero (married people), the predicted value is 30.2, the mean for married people. When X=1 (single), the predicted value is 24.8, the mean for single people. The b weight is equal to the difference in means between the groups. When we code the other way (1=married), the equation is Y' = 24.8+5.4X. Now when X is zero (single), the predicted value is 24.8, the mean of the single group. When X=1, the predicted value is 30.2, the mean of the married group.
4. The regression results are the same as what we got using ANOVA formulas for F and for t.

We can apply dummy coding to categorical variables with more than two levels. We can keep the use of zeros and ones as well. However, we will always need as many columns as there are degrees of freedom. With two levels, we need one column; with three levels, we need two columns. With C levels, we need C-1 columns.

Suppose we have three groups of people, single, married, and divorced, and we want to estimate their life satisfaction. Note how the first vector selects (identifies) the single group, and the second identifies the married group. The divorced folks are left over. The overall results will be the same, however, no matter which groups we select.

 Group Satis-faction Vector1 Vector2 Satis Group Mean Single 25 1 0 24.80 S 28 1 0 S 20 1 0 S 26 1 0 S 25 1 0 Married 30 0 1 30.20 M 28 0 1 M 32 0 1 M 33 0 1 M 28 0 1 Divorced 20 0 0 23.80 D 22 0 0 D 28 0 0 D 25 0 0 D 24 0 0 Grand Mean 26.27 .33 .33

The descriptive statistics for the variables are:

 Sat V1 V2 Satisfaction 1 Vector 1 -.28 1 Vector 2 .74 -.50 1 Mean 26.27 .33 .33 SD 3.88 .49 .49

When we run the program with satisfaction as the DV and the two vectors as the IVs, we find that R2 is .5619. The significance of this is found by:

Note that there are three groups and thus two degrees of freedom between groups. There are 15 people and thus 12 df for error. The F test based on R2 gives us the same result we would get if we used the traditional ANOVA approach to analyze these data.

The parameter estimates for these data are:

 Variable df Est Std Err t P > |t| Intercept 23.8 1.24 19.18 .0001 V1 1 1 1.75 .57 .5793 V2 1 6.4 1.75 3.65 .0033

Thus, the regression equation using this particular dummy code is:

Y' = 23.8 + 1(V1) + 6.4(V2)

Points to notice:

1. The single group is identified when X1 is 1 and X2 is zero. The married group is identified when X2 is 1 and X1 is zero. The divorced group is identified when both X1 and X2 are zero.
2. The intercept is 23.80. This is equal to the mean of the divorced group. The divorced group is identified when both X1 and X2 are zero, so they drop out of the regression equation, leaving our predicted value equal to the mean of the divorced group, as it should.
3. The value of b1 is 1. The mean of the single group is group is 24.80, which is 1 larger than the divorced group. So when X1 is 1 and X2 is zero, we identify the single group, and our predicted value is 23.80+1 = 24.80, the mean of the single group.
4. The value of b2 is 6.40. The mean of the married group is 30.20, which is 6.40 greater than that of the divorced group. So when X1 is zero and X2 is one, we identify the married group, and our predicted value is 23.80+ 6.40 = 30.20, the mean of the married group.

The group that gets all zeros is the base group or comparison group. The regression coefficients present a contrast or difference between the group identified by the vector and the base or comparison group. For our example, the comparison group is the divorced group. The first b weight corresponds to the single group and the b represents the difference between the means of the divorced and single groups. The second b weight represents the difference in means between the divorced and married groups.

The tests of significance of the b weights are equivalent to t-tests of the difference between the means of the identified and comparison groups.

Effect Coding

Effect coding is similar to dummy coding. The difference in coding is that, in effect coding, the comparison group is identified by the symbol -1. Our example looks like this:

 Group Satisfaction Vector1 Vector2 Satis Group Mean Single 25 1 0 24.80 S 28 1 0 S 20 1 0 S 26 1 0 S 25 1 0 Married 30 0 1 30.20 M 28 0 1 M 32 0 1 M 33 0 1 M 28 0 1 Divorced 20 -1 -1 23.80 D 22 -1 -1 D 28 -1 -1 D 25 -1 -1 D 24 -1 -1 Grand Mean 26.27 0 0 Descriptives Sat V1 V2 Satisfaction 1 Vector 1 .11 1 Vector 2 .70 .50 1 Mean 26.27 .0 .0 SD 3.88 .85 .85

The R2 for this model is also .5619. The estimates are somewhat different, however.

 Variable Df Est Std Err t P > |t| Intercept 26.27 .72 36.66 .0001 V1 1 -1.47 1.01 -1.45 .17 V2 1 3.93 1.01 3.88 .002

Note that regression equation is different.

Y' = 26.27 -1.47(V1)+3.93(V2)

Points to notice:

1. The intercept (a) is now the grand mean of satisfaction.
2. The b weights are now such that they specify the deviation of the identified group from the grand mean. For example, when V1 is 1 and V2 is zero (single group), the predicted Y value is 26.27-1.47 or 24.8, which is the mean of the single group.
3. Now the significance of the b weights tells whether the group differs significantly from the grand mean rather than from a chosen cell mean.
4. The deviation of a cell from the grand mean is known as a treatment effect. Hence the name effect coding.
5. It is also possible to use effect coding to conduct post hoc tests of differences among the means. We can also use it for planned comparisons. I'm not going to show you how to do these, however. Use orthogonal coding instead.

Orthogonal coding

Orthogonal coding is used to compute contrasts. You can use it if you have specific planned comparisons going into the analysis. Our example:

 Group Satisfaction Vector1 Vector2 Satis Mean Single 25 -1 1 24.80 S 28 -1 1 S 20 -1 1 S 26 -1 1 S 25 -1 1 Married 30 1 1 30.20 M 28 1 1 M 32 1 1 M 33 1 1 M 28 1 1 Divorced 20 0 -2 23.80 D 22 0 -2 D 28 0 -2 D 25 0 -2 D 24 0 -2 Grand Mean 26.27 0 0 Descriptives Sat V1 V2 Satisfaction 1 Vector 1 .59 1 Vector 2 .47 .00 1 Mean 26.27 0.0 0.0 SD 3.88 .85 1.46

Take a look at the contrasts implied by the positive and negative numbers in the two vectors. In the first vector, we are comparing single and married people; we are ignoring divorced people. In the second vector, we are comparing the mean of both single and married people to the mean of the divorced group. Notice in the correlation matrix that V1 and V2 are not correlated, hence orthogonal coding. There are only as many orthogonal contrasts allowed in one analysis as there are degrees of freedom. In this case, there are two. Exactly which two are tested depends entirely upon the design and hypothesized effects. In our example, we would need to specify in advance that we expect differences between single and married and between both these and divorced. We could have chosen other contrasts -- we could have hypothesized that the single would be different from the combined married and divorced, for example.

The R2 for this analysis is .5619, just as for dummy and effect coding.

 Variable df Est Std Err t P > |t| Intercept 26.27 .72 36.66 .0001 V1 1 2.70 .88 3.08 .01 V2 1 1.23 .51 2.43 .03

Points to notice:

1. The intercept is equal to the grand mean.
2. The b weights represent contrasts. The significance of the b weights represents the significance of the contrast. Our first contrast says that the means for the single and married groups differ. Our second test implies that the mean for the divorce is different from the combined mean of single and married.
3. The b weights themselves do not refer to straightforward mean differences between cells (as in dummy or effect coding) because of the coding of the IVs. For example, people in the single group have predicted scores of

Y' = 26.27 + 2.70(-1) + 1.23(1).

People in the divorced group have predicted scores of

Y' = 26.27 + 2.70(0) + 1.23(-2).

Unequal Sample Sizes

Designs in which the cells contain unequal frequencies introduce minor complications to the types of coding shown here. For effect coding, the meaning of the intercept changes. For effect coding, the intercept will no longer refer to the grand mean. Instead, it will denote the unweighted mean of the cell means.

where Yi is the individual scores on the DV, we have, say,

with three cells. Note that the results will not be the same if the cells do not have equal frequencies. In the case of equal frequencies, both values of mean Y, weighted and unweighted, are the same. With orthogonal coding, the intercept will still be the grand mean, but we have to change the values of the codes to maintain orthogonality. For dummy coding, the intercept still refers to the mean of the base or comparison group.

For orthogonal coding, you have to adjust the code numbers to keep the sums equal to zero and the vectors orthogonal. For example,

 Group Satisfaction Vector1 Vector2 Satis Group Mean Single 25 -4 5 24.33 S 28 -4 5 S 20 -4 5 Married 30 3 5 30.75 M 32 3 5 M 33 3 5 M 28 3 5 Divorced 20 0 -7 23.80 D 22 0 -7 D 28 0 -7 D 25 0 -7 D 24 0 -7 M 26.27 0 0

The correlation between V1 and V2 is still zero.

Otherwise, the results of the regressions are the same. All three types of codings give the same R2. The interpretation of the b weights is what it was before (for dummy coding, the contrast between a cell and a comparison cell, for effect, the contrast between a cell and the (unweighted) mean, and for orthogonal, specific planned comparisons.