### SAS LibraryHow do I handle interactions of continuous and categorical variables?

#### 1. A standard ANOVA2. A standard ANCOVA 3. Estimate slopes for each diet group4. Test equality of slopes across diet groups5. Perform tests with separate slopes for all diet groups5.1 Comparing diet 1 with diet 2      5.2 Comparing diets 1 and 2 to the control group 6. Testing to pool slopes7. Perform tests with some pooled slopes 7.1 Overall analysis pooling slopes for diet groups 2 and 3   7.2 Comparing diet groups 1 and 2 when pooling slopes for diet groups 2 and 3   7.3 Comparing diet groups 2 and 3 when pooling slopes for diet groups 2 and 3   8. Summary

Analysis of covariance (ANCOVA) is a statistical procedure that allows you to include both categorical and continuous variables in a single model. ANCOVA assumes that the regression coefficients are homogeneous (the same) across the categorical variable. Violation of this assumption can lead to incorrect conclusions. This page will explore what happens when you have heterogeneous (different) regressions across groups and show some strategies for dealing with them. This involves some complex topics in the use of proc glm, especially the estimate statement.

Here is an example data file we will use. It contains 30 subjects who used one of three diets, diet 1 (diet=1), diet 2 (diet=2) and a control group (diet=3). Before the start of the study, the height of the subject was measured, and after the study the weight of the subject was measured.

DATA htwt;
INPUT id diet height weight ;
CARDS;
1 1  56 140
2 1  60 155
3 1  64 143
4 1  68 161
5 1  72 139
6 1  54 159
7 1  62 138
8 1  65 121
9 1  65 161
10 1  70 145
11 2  56 117
12 2  60 125
13 2  64 133
14 2  68 141
15 2  72 149
16 2  54 109
17 2  62 128
18 2  65 131
19 2  65 131
20 2  70 145
21 3  54 211
22 3  58 223
23 3  62 235
24 3  66 247
25 3  70 259
26 3  52 201
27 3  59 228
28 3  64 245
29 3  65 241
30 3  72 269
;
RUN;

#### 1. A standard ANOVA

You could analyze this data with a standard ANOVA, as shown below. This analysis compares the weights of the three groups. It also uses the contrast statement to compare the two diets (1 and 2) to the control group (diet 3).  We also want to compare diet 1 with diet 2.
PROC GLM DATA=htwt;
CLASS diet ;
MODEL weight = diet ;
MEANS diet / deponly ;
CONTRAST 'compare 1&2 with control' diet  1  1 -2 ;
CONTRAST 'compare diet 1 with 2   ' diet  1 -1  0 ;
RUN;
QUIT;
General Linear Models Procedure
Class Level Information

Class    Levels    Values

DIET          3    1 2 3

Number of observations in data set = 30

General Linear Models Procedure

Dependent Variable: WEIGHT
Sum of            Mean
Source                  DF          Squares          Square   F Value     Pr > F

Model                    2     64350.600000    32175.300000    128.48     0.0001

Error                   27      6761.400000      250.422222

Corrected Total         29     71112.000000

R-Square             C.V.        Root MSE          WEIGHT Mean

0.904919         9.254231       15.824735            171.00000

Source                  DF        Type I SS     Mean Square   F Value     Pr > F

DIET                     2     64350.600000    32175.300000    128.48     0.0001

Source                  DF      Type III SS     Mean Square   F Value     Pr > F

DIET                     2     64350.600000    32175.300000    128.48     0.0001

General Linear Models Procedure

Level of        ------------WEIGHT-----------
DIET        N       Mean              SD

1          10     146.200000       12.8391762
2          10     130.900000       12.2424580
3          10     235.900000       20.8936460
General Linear Models Procedure

Dependent Variable: WEIGHT

Contrast                DF      Contrast SS     Mean Square   F Value     Pr > F

compare 1&2 with con     1     63180.150000    63180.150000    252.29     0.0001
compare diet 1 with      1      1170.450000     1170.450000      4.67     0.0397


The ANOVA results show an overall difference among all of the diets and the contrasts show a difference between the control group and the two diets, and a difference between diet 1 and diet 2.   The ANOVA disregards the information that we have about the subject's height. As height is probably correlated with weight, this could be useful as a covariate in an ANCOVA.

2. A standard ANCOVA

Below we perform a standard ANCOVA.
PROC GLM DATA=htwt;
CLASS diet ;
MODEL weight = diet height / SOLUTION ;
LSMEANS diet ;
CONTRAST 'compare control with 1,2' diet  1  1 -2 ;
CONTRAST 'compare 1 with 2        ' diet  1 -1  0 ;
RUN;
QUIT;
The results are consistent with those of the ANOVA. There is an overall effect of diet. Also, the control group is significantly different from the two diets, and diet 1 is different from diet 2. The significance level for the comparison of diet 1 versus diet 2 is smaller than the standard ANOVA.
General Linear Models Procedure
Class Level Information

Class    Levels    Values

DIET          3    1 2 3

Number of observations in data set = 30
General Linear Models Procedure

Dependent Variable: WEIGHT
Sum of            Mean
Source                  DF          Squares          Square   F Value     Pr > F

Model                    3     67409.811075    22469.937025    157.80     0.0001

Error                   26      3702.188925      142.391882

Corrected Total         29     71112.000000

R-Square             C.V.        Root MSE          WEIGHT Mean

0.947939         6.978250       11.932807            171.00000

Source                  DF        Type I SS     Mean Square   F Value     Pr > F

DIET                     2     64350.600000    32175.300000    225.96     0.0001
HEIGHT                   1      3059.211075     3059.211075     21.48     0.0001

Source                  DF      Type III SS     Mean Square   F Value     Pr > F

DIET                     2     66726.086524    33363.043262    234.30     0.0001
HEIGHT                   1      3059.211075     3059.211075     21.48     0.0001

T for H0:    Pr > |T|   Std Error of
Parameter                  Estimate    Parameter=0                Estimate

INTERCEPT               126.1382736 B         5.26     0.0001    23.97915732
DIET      1             -92.1705212 B       -17.19     0.0001     5.36306483
2            -107.4705212 B       -20.04     0.0001     5.36306483
3               0.0000000 B          .        .          .
HEIGHT                    1.7646580           4.64     0.0001     0.38071364

NOTE: The X'X matrix has been found to be singular and a generalized inverse
was used to solve the normal equations.   Estimates followed by the
letter 'B' are biased, and are not unique estimators of the parameters.

General Linear Models Procedure
Least Squares Means

DIET        WEIGHT
LSMEAN

1       145.376493
2       130.076493
3       237.547014

General Linear Models Procedure

Dependent Variable: WEIGHT

Contrast                DF      Contrast SS     Mean Square   F Value     Pr > F

compare control with     1     65555.636524    65555.636524    460.39     0.0001
compare 1 with 2         1      1170.450000     1170.450000      8.22     0.0081


Because we used the solution option, we are shown the regression coefficients and see the coefficient (slope) between height and weight is 1.76. Figure 1 below shows the scatterplot between height and weight and the line of best fit with slope 1.76.

Figure 1. Scatterplot of weight by height with overall regression line

3. Estimate slopes for each diet group

One assumption of ANCOVA is that the slope between height and weight is the same for the three diet groups. This is called the homogeneity of regression assumption. Below we show a scatterplot like the one above; however, this one shows the three diet groups in different colors and shows a separate regression line for each diet group (diet 1=blue, diet 2=yellow, diet 3=red). As you can see the blue regression line looks like it has a very different slope from the other two regression lines.

Figure 2. Scatterplot of weight by height with separate regression lines for each group (diet 1=blue, diet 2=yellow, diet 3=red)

Below we perform an analysis that shows the slopes of each of the lines. Even if we found the slope between height and weight to be 0 in the prior analysis, this is still a useful analysis to perform. It is possible that the overall slope for the entire sample was 0, but the slopes for some groups were positive and the others were negative and they cancelled each other out. This analysis would help you see if such a pattern was occurring.
PROC SORT DATA=htwt;
BY diet;
RUN;

PROC GLM DATA=htwt;
BY diet ;
MODEL weight = height / SOLUTION ;
RUN;
QUIT;
We indeed see below that the slopes seem very different. (Note that the output has been abbreviated.)  The slope for diet 1 (-.37) is much smaller than the slope for diet 2 (2.095) and the control group, diet=3 (3.189).  We need to check into this further and test whether these slopes are significantly different from each other.
diet=1

Dependent Variable: weight
Standard
Parameter         Estimate           Error    t Value    Pr > |t|

Intercept      170.1664447     49.43018216       3.44      0.0088
height          -0.3768309      0.77433413      -0.49      0.6396

diet=2

Dependent Variable: weight

Standard
Parameter         Estimate           Error    t Value    Pr > |t|

Intercept     -2.397470040      7.05327189      -0.34      0.7427
height         2.095872170      0.11049098      18.97      <.0001

diet=3

Dependent Variable: weight

Standard
Parameter         Estimate           Error    t Value    Pr > |t|

Intercept      37.49895178      7.70303233       4.87      0.0012
height          3.18972746      0.12323669      25.88      <.0001

4. Test equality of slopes across diet groups

We can test to see if the slopes for the three diet groups are equal, as shown below. The diet*height effect tests if the three slopes are equal.
PROC GLM DATA=htwt;
CLASS diet ;
MODEL weight = diet height diet*height ;
RUN;
QUIT;
The diet*height effect is indeed significant, indicating that the slopes do differ across the three diet groups. (Note that we look at the Type III SS, consistent with our general recommendations to use Type III instead of Type I SS.) The output is abbreviated to save space.
<some output omitted>

Source                  DF      Type III SS     Mean Square   F Value     Pr > F
DIET                     2     1224.0043440     612.0021720      9.68     0.0008
HEIGHT                   1     2597.0189017    2597.0189017     41.10     0.0001
HEIGHT*DIET              2     2185.5435689    1092.7717845     17.29     0.0001


5. Perform tests with separate slopes for all diet groups

Because the slopes for the three diet groups are not the same, we should not use a traditional ANCOVA model that assumes the slopes for the three diet groups are the same.  Instead, we can use a model that estimates separate slopes for all three diet groups. Because the diet groups will have different slopes, we must be very cautious in interpreting adjusted means. One way of thinking about this is to focus on the fact that we have a diet*height interaction. This means that we cannot interpret the relationship between height and weight without referring to diet. Likewise, if we want to talk about the effect of diet we need to specify what height we are talking about. For example, in comparing diets 1 and 2 (in Figure 2) it looks like there is no difference between diets 1 and 2 (blue and yellow) for tall people, but there may be a difference for shorter people. Below, we will see how to make these comparisons.

Let us compare diet 1 versus diet 2 at three different levels of height, for those who are 59 inches tall, 64 inches and 68 inches tall. These correspond to the 25th, 50th and 75th percentiles for height. We can then evaluate separately for each height group the difference between diet 1 and diet 2. The model used in this analysis is the same as the model from section 4 where we estimated separate slopes. In addition we use the estimate statement for comparing the diets 1 and 2 at the three levels of height, and for obtaining the adjusted mean for weight.

The first three estimate statements compare diet 1 with diet 2 at 59, 64, and 68 inches. The next three estimate statements request the predicted value of weight for people on diet 1 who are 59 inches, 64 inches, and 68 inches tall. The next three estimate statements requests the weight for people on diet 2 who are 59 inches, 64 inches, and 68 inches tall.

PROC GLM DATA=htwt;
CLASS diet ;
MODEL weight = diet height diet*height ;

ESTIMATE 'diet 1 vs 2 at 59 in' diet -1 1 0 diet*height -59 59 0 ;
ESTIMATE 'diet 1 vs 2 at 64 in' diet -1 1 0 diet*height -64 64 0 ;
ESTIMATE 'diet 1 vs 2 at 68 in' diet -1 1 0 diet*height -68 68 0 ;

ESTIMATE 'wt for diet 1 at 59 in' intercept 1 diet 1 0 0 height 59 diet*height 59 0 0 ;
ESTIMATE 'wt for diet 1 at 64 in' intercept 1 diet 1 0 0 height 64 diet*height 64 0 0 ;
ESTIMATE 'wt for diet 1 at 68 in' intercept 1 diet 1 0 0 height 68 diet*height 68 0 0 ;

ESTIMATE 'wt for diet 2 at 59 in' intercept 1 diet 0 1 0 height 59 diet*height 0 59 0 ;
ESTIMATE 'wt for diet 2 at 64 in' intercept 1 diet 0 1 0 height 64 diet*height 0 64 0 ;
ESTIMATE 'wt for diet 2 at 68 in' intercept 1 diet 0 1 0 height 68 diet*height 0 68 0 ;

RUN;
QUIT;
For the sake of saving space, we show just the output related to the estimate statements.
<output omitted to save space>
Standard
Parameter                     Estimate           Error    t Value    Pr > |t|

diet 1 vs 2 at 59 in        -26.674434      4.64126551      -5.75      <.0001
diet 1 vs 2 at 64 in        -14.310919      3.56455159      -4.01      0.0005
diet 1 vs 2 at 68 in         -4.420107      4.55895082      -0.97      0.3419
wt for diet 1 at 59 in      147.933422      3.28187031      45.08      <.0001
wt for diet 1 at 64 in      146.049268      2.52051860      57.94      <.0001
wt for diet 1 at 68 in      144.541944      3.22366504      44.84      <.0001
wt for diet 2 at 59 in      121.258988      3.28187031      36.95      <.0001
wt for diet 2 at 64 in      131.738349      2.52051860      52.27      <.0001
wt for diet 2 at 68 in      140.121838      3.22366504      43.47      <.0001

Focusing on the comparison of diets 1 and 2, these results indicate a significant difference between diet 1 and diet 2 for those 59 inches tall (t=-5.75, p < .0001) and a significant difference for those 64 inches tall (t=-4.01, p=0.0005).  For those who are tall (i.e., 68 inches), diet 1 and diet 2 are about equally effective.  This corresponds with what we saw in Figure 2.

You will notice that if you take the parameter estimate for "wt for diet 1 at 59 in" minus the parameter estimate for "wt for diet 2 at 59 in", you get -26.67, which is the parameter estimate for "diet 1 vs. 2 at 59 in" (147.93 - 121.25 = -26.67). Likewise, taking the parameter estimate for "wt for diet 1 at 64 in" minus the parameter estimate for "wt for diet 2 at 64in" yields the parameter estimate for "diet 1 vs. 2 at 64 in" (146.04-131.73 = -14.31).  You can do a similar computation for the weights for those 68 inches tall.

It can be very difficult to construct estimate statements, so here is a shortcut that works when you want to compare one group versus another group using the lsmeans statement.
PROC GLM DATA=htwt;
CLASS diet ;
MODEL weight = diet height diet*height  ;
LSMEANS diet / AT height=59 PDIFF ;
LSMEANS diet / AT height=64 PDIFF ;
LSMEANS diet / AT height=68 PDIFF ;
RUN;
QUIT;

For the sake of saving space, we show just the output related to the lsmeans statements. As you can see, the results comparing diet 1 and 2 are the same using lsmeans as using estimate. For example, for those 59 inches tall, the adjusted mean for diet 1 is 147.93 and the adjusted mean for diet 2 is 121.25, and this is significant with p<.0001 (see italics in output).

<some output omitted to save space>

The GLM Procedure
Least Squares Means at height=59

weight      LSMEAN
diet          LSMEAN      Number

1         147.933422           1
2         121.258988           2
3         225.692872           3

Least Squares Means for effect diet
Pr > |t| for H0: LSMean(i)=LSMean(j)

Dependent Variable: weight

i/j              1             2             3

1                      <.0001        <.0001
2        <.0001                      <.0001
3        <.0001        <.0001

NOTE: To ensure overall protection level, only probabilities associated with pre-planned
comparisons should be used.
The GLM Procedure
Least Squares Means at height=64

weight      LSMEAN
diet          LSMEAN      Number

1         146.049268           1
2         131.738349           2
3         241.641509           3

Least Squares Means for effect diet
Pr > |t| for H0: LSMean(i)=LSMean(j)

Dependent Variable: weight

i/j              1             2             3

1                      0.0005        <.0001
2        0.0005                      <.0001
3        <.0001        <.0001

NOTE: To ensure overall protection level, only probabilities associated with pre-planned
comparisons should be used.
The GLM Procedure
Least Squares Means at height=68

weight      LSMEAN
diet          LSMEAN      Number

1         144.541944           1
2         140.121838           2
3         254.400419           3

Least Squares Means for effect diet
Pr > |t| for H0: LSMean(i)=LSMean(j)

Dependent Variable: weight

i/j              1             2             3

1                      0.3419        <.0001
2        0.3419                      <.0001
3        <.0001        <.0001

NOTE: To ensure overall protection level, only probabilities associated with pre-planned
comparisons should be used.


It each much easier to use lsmeans for comparing groups than estimate, however lsmeans can only make pairwise comparisons. But we are also interested in comparing the two diets (diet 1 and diet 2) combined to the control group (diet 3). For that, we must use estimate since lsmeans will not do that kind of comparison, as shown below.

The analysis below compares diets 1 and 2 to the control group (group 3) at the three different heights: 59 inches, 64 inches and 68 inches. The first three estimate commands compare diets 1 and 2 to the control group at these three different heights. The next three estimate commands estimate the weight for the diet 1 and diet 2 groups combined at the three heights.  The following estimate commands estimate the weight for the control group at the three heights.

PROC GLM DATA=htwt;
CLASS diet ;
MODEL weight = diet height diet*height  ;

ESTIMATE 'diet 1&2 vs 3 at 59in' diet .5 .5 -1  diet*height 29.5 29.5 -59 ;
ESTIMATE 'diet 1&2 vs 3 at 64in' diet .5 .5 -1  diet*height 32   32   -64 ;
ESTIMATE 'diet 1&2 vs 3 at 68in' diet .5 .5 -1  diet*height 34   34   -68 ;

ESTIMATE 'wt diet 1&2 at 59in' intercept 1 diet .5 .5 0 height 59 diet*height 29.5 29.5 0 ;
ESTIMATE 'wt diet 1&2 at 64in' intercept 1 diet .5 .5 0 height 64 diet*height 32   32   0 ;
ESTIMATE 'wt diet 1&2 at 68in' intercept 1 diet .5 .5 0 height 68 diet*height 34   34   0 ;

ESTIMATE 'wt control at 59in'  intercept 1 diet 0 0 1 height 59 diet*height 0 0 59 ;
ESTIMATE 'wt control at 64in'  intercept 1 diet 0 0 1 height 64 diet*height 0 0 64 ;
ESTIMATE 'wt control at 68in'  intercept 1 diet 0 0 1 height 68 diet*height 0 0 68 ;

RUN;
QUIT;

For the sake of saving space, we show just the output related to the estimate statements.

<some output omitted to save space>
Standard
Parameter                    Estimate           Error    t Value    Pr > |t|

diet 1&2 vs 3 at 59in      -91.096667      3.66066279     -24.89      <.0001
diet 1&2 vs 3 at 64in     -102.747701      3.16739826     -32.44      <.0001
diet 1&2 vs 3 at 68in     -112.068528      4.13354574     -27.11      <.0001
wt diet 1&2 at 59in        134.596205      2.32063275      58.00      <.0001
wt diet 1&2 at 64in        138.893808      1.78227579      77.93      <.0001
wt diet 1&2 at 68in        142.331891      2.27947541      62.44      <.0001
wt control at 59in         225.692872      2.83109796      79.72      <.0001
wt control at 64in         241.641509      2.61837826      92.29      <.0001
wt control at 68in         254.400419      3.44821580      73.78      <.0001


The output indicates the difference in weight between diet groups 1 and 2 combined and the control group is -91.0967 pounds at 59 inches, and this difference is significant. We could obtain that difference by taking 134.59 (the average for diet groups 1 and 2 at 59 inches) minus 225.69 (the average for diet group 3 at 59 inches). Likewise, the difference between diet groups 1 and 2 versus diet group 3 is significant at 64 inches (with a difference of -102.74 pounds) and at 68 inches (with a difference of -112.069 pounds). Despite the interaction, the control group (diet 3) always weighs more than the two diet groups combined. This is consistent with what we saw in figure 2.

6. Testing to pool slopes

You may have noticed that the slope for diet group 1 was quite different from 2 and 3, but 2 and 3 were not so different from each other (see the graph from figure 2 and output in section 4)  Rather than estimating three separate slopes, maybe it would be better if we estimated a slope for diet group 1, and one combined slope for diet groups 2 and 3. Let's compare the slopes for diet groups 2 and 3 to see if they are different (and if they are not different they can be combined), and also test to see if the slope for diet group 1 is really different from the combined slopes for diet groups 2 and 3.
PROC GLM DATA=htwt;
CLASS diet ;
MODEL weight = diet height  diet*height  ;
ESTIMATE 'compare 1 vs 2&3' diet*height -2  1  1 ;
ESTIMATE 'compare 2 vs 3'   diet*height  0 -1  1 ;
RUN;
QUIT;

For the sake of saving space, we show just the output related to the estimate statements.

<some output omitted to save space>

Parameter                   Estimate           Error    t Value    Pr > |t|

compare 1 vs 2&3          6.03926142      1.10336987       5.47      <.0001
compare 2 vs 3            1.09385529      0.61316088       1.78      0.0871

As we expected, the test comparing the slopes of diet group 1 versus 2 and 3 was significant, and the test comparing the slopes for diet groups 2 versus 3 was not significant. Because the slopes for diet groups 2 and 3 do not significantly differ, we can simplify our model by including one slope for diet group 1, and one combined slope for diet groups 2 and 3. This model has two benefits: 1) The estimate of the slope for diet groups 2 and 3 will be more stable (because it is based on more cases) than slopes computed separately. Second, as we will see later, comparisons between diet groups 2 and 3 are greatly simplified since they will have a common slope.

7. Perform tests with some pooled slopes

7.1 Overall analysis pooling slopes for diet groups 2 and 3

Let's see how we can can estimate a model with one slope for diet group 1, and another slope for diet groups 2 and 3. First, we will make a dummy variable that is 0 for diet group 1, and 1 for diet groups 2 and 3, called diet23.

data htwt2;
set htwt;

if diet = 1       then diet23 = 0;
if diet in (2,3)  then diet23 = 1;
run;

PROC FREQ DATA=htwt2;
TABLES diet*diet23 / LIST MISSING ;
RUN;
The diet23 variable has been created successfully.
                                   Cumulative  Cumulative
DIET  DIET23   Frequency  Percent   Frequency    Percent
---------------------------------------------------------
1       0          10    33.3           10       33.3
2       1          10    33.3           20       66.7
3       1          10    33.3           30      100.0


Now, we can use diet23 in our model. Both diet and diet23 are included as class variables. The variable diet is included in the model statement to indicate the mean differences among the three different diet groups, and diet23*height is used to indicate that we want to estimate two slopes.

PROC GLM DATA=htwt2;
CLASS diet diet23 ;
MODEL weight = diet height diet23*height ;
RUN;
QUIT;
Notice that diet has 2 df (since it has three levels) but the interaction of diet23*height has only 1 df (since diet23 has only two levels), whereas in section 4 the diet*height interaction had 2 df (since diet has three levels).
General Linear Models Procedure
Class Level Information

Class    Levels    Values
DIET          3    1 2 3

DIET23        2    0 1

Number of observations in data set = 30
The GLM Procedure

Dependent Variable: weight

Sum of
Source                      DF         Squares     Mean Square    F Value    Pr > F

Model                        4     69394.24001     17348.56000     252.49    <.0001

Error                       25      1717.75999        68.71040

Corrected Total             29     71112.00000

R-Square     Coeff Var      Root MSE    weight Mean

0.975844      4.847470      8.289174       171.0000

Source                      DF       Type I SS     Mean Square    F Value    Pr > F

diet                         2     64350.60000     32175.30000     468.27    <.0001
height                       1      3059.21107      3059.21107      44.52    <.0001
height*diet23                1      1984.42894      1984.42894      28.88    <.0001

Source                      DF     Type III SS     Mean Square    F Value    Pr > F

diet                         2     58738.39848     29369.19924     427.43    <.0001
height                       1      1133.21656      1133.21656      16.49    0.0004
height*diet23                1      1984.42894      1984.42894      28.88    <.0001


7.2 Comparing diet groups 1 and 2 when pooling slopes for diet groups 2 and 3

Even though we have pooled the slopes for groups 2 and 3, when we want to compare groups 1 and 2 we are comparing across groups with different slopes so we still need to use estimate to compare the diets at the different levels of heights and obtain the adjusted means. The first three estimate statements below compare diet groups 1 with 2 at the three levels of height (59, 64 and 68 inches). The next three estimate statements obtain adjusted means for diet 1 at the three heights, and the next three estimate statements obtain adjusted means for diet 2 at the three heights.

PROC GLM DATA=htwt2;
CLASS diet diet23 ;
MODEL weight = diet height diet23*height  ;

ESTIMATE 'diet 1 vs 2 at 59in' diet 1 -1 0 diet23*height  59 -59 ;
ESTIMATE 'diet 1 vs 2 at 64in' diet 1 -1 0 diet23*height  64 -64 ;
ESTIMATE 'diet 1 vs 2 at 68in' diet 1 -1 0 diet23*height  68 -68 ;

ESTIMATE 'wt for diet 1 at 59in' intercept 1 diet 1 0 0 height 59 diet23*height 59  0 ;
ESTIMATE 'wt for diet 1 at 64in' intercept 1 diet 1 0 0 height 64 diet23*height 64  0 ;
ESTIMATE 'wt for diet 1 at 68in' intercept 1 diet 1 0 0 height 68 diet23*height 68  0 ;

ESTIMATE 'wt for diet 2 at 59in' intercept 1 diet 0 1 0 height 59 diet23*height  0 59 ;
ESTIMATE 'wt for diet 2 at 64in' intercept 1 diet 0 1 0 height 64 diet23*height  0 64 ;
ESTIMATE 'wt for diet 2 at 68in' intercept 1 diet 0 1 0 height 68 diet23*height  0 68 ;

RUN;
QUIT;

We have omitted the portion of the output that was the same as that in section 7.1 to save space.

<some output omitted to save space>
Standard
Parameter                    Estimate           Error    t Value    Pr > |t|

diet 1 vs 2 at 59in         29.489844      4.55124555       6.48      <.0001
diet 1 vs 2 at 64in         14.066100      3.71413467       3.79      0.0009
diet 1 vs 2 at 68in          1.727105      4.48561874       0.39      0.7035
wt for diet 1 at 59in      147.933422      3.42212803      43.23      <.0001
wt for diet 1 at 64in      146.049268      2.62823833      55.57      <.0001
wt for diet 1 at 68in      144.541944      3.36143523      43.00      <.0001
wt for diet 2 at 59in      118.443578      3.00047926      39.47      <.0001
wt for diet 2 at 64in      131.983167      2.62433986      50.29      <.0001
wt for diet 2 at 68in      142.814839      2.97010584      48.08      <.0001

We can compare the results here with those of section 5.1 (which also compared groups 1 and 2, but estimated separate slopes for all three groups).  We see that the results are quite consistent, i.e., the difference between diet groups 1 and 2 are different at 59 inches, 64 inches, but not at 68 inches.

Because we have estimated a common slope for diet groups 2 and 3, it is easier to compare diet groups 2 and 3. Since the slopes for these two groups are parallel, we can compare these two groups at any value for height and the difference between the regression lines will remain constant. Hence, to compare diets 2 and 3, we only need diet 0 1 -1 in the estimate statement.  To obtain traditional adjusted means for each diet, you would estimate the adjusted mean at the overall mean value of height (in this case 63.13) as shown below.

PROC GLM DATA=htwt2;
CLASS diet diet23 ;
MODEL weight = diet height diet23*height ;
ESTIMATE 'diet 2 vs 3' diet 0 1 -1  ;
ESTIMATE 'diet 2 at xbar' intercept 1 diet 0 1 0 height 63.13 diet23*height 0 63.13 ;
ESTIMATE 'diet 3 at xbar' intercept 1 diet 0 0 1 height 63.13 diet23*height 0 63.13 ;
RUN;
QUIT;

We have omitted the portion of the output that was the same as that in section 7.1 to save space.  The comparison of diets 2 and 3 is significant, and this holds true across all levels of height.  Those in diet group 2 weighed about 108.8 pounds less than those in diet group 3.  For those of average height, the adjusted mean for diet 2 was 129.6 and for diet 3 was 238.4 (and 129.6 - 238.4 = -108.8).

                                            Standard
Parameter                   Estimate           Error    t Value    Pr > |t|
diet 2 vs 3              -108.791085      3.73357024     -29.14      <.0001
diet 2 at xbar            129.627279      2.62550857      49.37      <.0001
diet 3 at xbar            238.418364      2.63783571      90.38      <.0001

8. Summary

We have seen that in ANCOVA it is important to test the homogeneity of regression assumption, and if this assumption is violated we then need to estimate models that have separate slopes across groups.  This amounts to having an interaction between your covariate and your group variable, which means that when you estimate differences among the groups, you need to take the level of the covariate into consideration.  One strategy, as illustrated here, is to look at the effect of your group variable at different levels of your covariate.  In our example, when we compared the control group to diets 1 and 2, we found that the control group weighed more at 3 different levels of height (59 inches, 64 inches and 68 inches).  However, when we compared diets 1 and 2, we found diet 2 to be more effective at 59 and 64 inches, but there was no difference at 68 inches.  Had we not done this further investigation, we may have concluded that diet 1 was superior to diet 2 for people of all heights, not realizing that the effectiveness of the diet depended on height.

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