ED230B/C

Analysis of Covariance

Four Possibilities

Common Slopes - Common Intercepts

regress write read
[output omitted]

predict p1

graph write p1 read, s(oi) c(.L) sort            /* stata 7 */
scatter write p1 read, msym(o i) con(. L) sort   /* stata 8 */

Common Slopes - Different Intercepts

regress write read female
[output omitted]

predict p2
sort female p2
  
graph write p2 read, s(oi) c(.L)                 /* stata 7 */
scatter write p2 read, msym(o i) con(. L) sort   /* stata 8 */

Different Slopes - Different Intercepts

generate fxr = female*read

regress write read female fxr
[output omitted]

predict p3
sort female p3

graph write p3 read, s(oi) c(.L) sort            /* stata 7 */
scatter write p3 read, msym(o i) con(. L) sort   /* stata 8 */

Different Slope - Common Intercepts

Classical ANCOVA

Partial out effects of one or more nuisance variables from the dependent variable.
Random assignment to groups.
One or more continuous variables.
Statistical control vs experimental control.

Assumptions

Independence
Normality
Homogeneity of variance
Population within group regression coefficients are equal
covariate is measured without error
Regression residuals are NID with mean 0 and equal variances
Relationship between covariate & dependent variable is linear
Covariate is related to dependent variable but is logically independent of treatment.

Selecting a Covariate

The experiment contains one or more extraneous sources of variation--
- Related to the dependent variable
- Irrelevant to the independent variable
Experimental control is not feasible or possible
If possible measure covariate so that it does not include treatment effects
- Covariate obtained prior to treatment
- Covariate obtained after treatment but before it take effect
- Covariate is assumed to be unaffected by treatment

Logic of ANCOVA

Which may be rewritten:

Homogeneity of Regression

The previous two formulas involved the use of a common regression coefficient, b.
Assumes that the slope of the regression line is constant across groups.
Test the interaction of the independent variable and the covariate.

Steps in ANCOVA

Include variable of scores on the dependent variable -- y
Include variables of scores on the covariates -- c
Create variables for coded group membership -- v
Multiply each v by c to create interaction vectors -- cv
Test variance added by step 4 -- homogeneity of regression slopes -- If n.s. procede to step 6; if sig. analyze as ATI
Test variance added by step 3 -- If sig. then there are treatment effects, procede to step 7; if n.s. calculate regression with covariate only.
If there are more than 2 levels perform multiple comparisons.

Numerical Example: coded using Effect Coding

input id  y  c1  c2 grp  v1  v2  v3
 1  6   1   6   1   1   0   0   
 2  9   1   7   1   1   0   0
 3  8   2  15   1   1   0   0
 4  8   3  13   1   1   0   0
 5 12   3  18   1   1   0   0
 6 12   4   9   1   1   0   0
 7 10   4  16   1   1   0   0
 8  8   5  10   1   1   0   0
 9 12   5  16   1   1   0   0
10 13   6  18   1   1   0   0   
11 13   4  12   2   0   1   0   
12 16   4  12   2   0   1   0
13 15   5  17   2   0   1   0
14 16   6   9   2   0   1   0
15 19   6  20   2   0   1   0
16 17   8  18   2   0   1   0
17 19   8  16   2   0   1   0
18 23   9  20   2   0   1   0
19 19  10  10   2   0   1   0
20 22  10  17   2   0   1   0    
21 20   7   8   3   0   0   1    
22 22   7  14   3   0   0   1
23 24   9  11   3   0   0   1
24 26   9  11   3   0   0   1
25 24  10  16   3   0   0   1
26 25  11  20   3   0   0   1
27 28  11  19   3   0   0   1
28 27  12  19   3   0   0   1
29 29  13  12   3   0   0   1
30 26  13  16   3   0   0   1   
31 27   7  16   4  -1  -1  -1  
32 28   8  10   4  -1  -1  -1
33 25   8  13   4  -1  -1  -1
34 27   9   7   4  -1  -1  -1
35 31   9  15   4  -1  -1  -1
36 29  10  20   4  -1  -1  -1
37 32  10  16   4  -1  -1  -1
38 30  12  21   4  -1  -1  -1
39 32  12  15   4  -1  -1  -1
40 33  14  21   4  -1  -1  -1  
end

/* compute interaction terms */
generate c1v1 = c1 * v1
generate c1v2 = c1 * v2
generate c1v3 = c1 * v3

/* run regression */
regress y c1 v1 v2 v3 c1v1 c1v2 c1v3

  Source |       SS       df       MS                  Number of obs =      40
---------+------------------------------               F(  7,    32) =  109.27
   Model |  2382.23359     7  340.319084               Prob > F      =  0.0000
Residual |  99.6664092    32  3.11457529               R-squared     =  0.9598
---------+------------------------------               Adj R-squared =  0.9511
   Total |     2481.90    39  63.6384615               Root MSE      =  1.7648

------------------------------------------------------------------------------
       y |      Coef.   Std. Err.       t     P>|t|       [95% Conf. Interval]
---------+--------------------------------------------------------------------
      c1 |   .9994664   .1429378      6.992   0.000       .7083116    1.290621
      v1 |  -6.107133   1.452131     -4.206   0.000      -9.065026    -3.14924
      v2 |  -2.671148   1.736811     -1.538   0.134      -6.208916    .8666195
      v3 |    1.54334   2.267783      0.681   0.501      -3.075983    6.162663
    c1v1 |  -.0979512   .2818144     -0.348   0.730      -.6719884     .476086
    c1v2 |   .1047003    .229945      0.455   0.652      -.3636823    .5730829
    c1v3 |   .0509923   .2369574      0.215   0.831      -.4316741    .5336588
   _cons |   12.84198   1.127409     11.391   0.000       10.54552    15.13844
------------------------------------------------------------------------------

/* test homogeneity of regression slopes */
test c1v1 c1v2 c1v2

 ( 1)  c1v1 = 0.0
 ( 2)  c1v2 = 0.0
 ( 3)  c1v3 = 0.0

       F(  3,    32) =    0.11
            Prob > F =    0.9550


/* run regression again */
regress y c1 v1 v2 v3

  Source |       SS       df       MS                  Number of obs =      40
---------+------------------------------               F(  4,    35) =  206.97
   Model |  2381.22741     4  595.306852               Prob > F      =  0.0000
Residual |  100.672592    35  2.87635976               R-squared     =  0.9594
---------+------------------------------               Adj R-squared =  0.9548
   Total |     2481.90    39  63.6384615               Root MSE      =   1.696

------------------------------------------------------------------------------
       y |      Coef.   Std. Err.       t     P>|t|       [95% Conf. Interval]
---------+--------------------------------------------------------------------
      c1 |   1.013052   .1337037      7.577   0.000       .7416185    1.284485
      v1 |  -6.469857   .7313256     -8.847   0.000      -7.954527   -4.985187
      v2 |  -2.016843   .4719217     -4.274   0.000      -2.974895   -1.058791
      v3 |   1.941392   .5781528      3.358   0.002       .7676797    3.115105
   _cons |   12.82548   1.054168     12.166   0.000       10.68541    14.96556
------------------------------------------------------------------------------

/* test treatment */
test v1 v2 v3

 ( 1)  v1 = 0.0
 ( 2)  v2 = 0.0
 ( 3)  v3 = 0.0

       F(  3,    35) =   48.19
            Prob > F =    0.0000

/* test covariate (redundant) */
test c1

 ( 1)  c1 = 0.0

       F(  1,    35) =   57.41
            Prob > F =    0.0000

/* compute original means */
table grp, contents(mean y)

----------+-----------
      grp |    mean(y)
----------+-----------
        1 |        9.8
        2 |       17.9
        3 |       25.1
        4 |       29.4
----------+-----------

/* compute adjusted means */
adjust c1, by(grp) generate(adjmeans)

-------------------------------------------------------------------------------
Dependent variable: y     Command: regress
Created variable: adjmeans
Variables left as is: v1, v2, v3
Covariate set to mean: c1 = 7.625
-------------------------------------------------------------------------------

----------+-----------
      grp |         xb
----------+-----------
        1 |    14.0801
        2 |    18.5332
        3 |    22.4914
        4 |    27.0953
----------+-----------
Key:  xb         =  Linear Prediction

Regression Equation

Separate Intercepts

Computing Adjusted Means

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UCLA Department of Education

Phil Ender, 22Feb00