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How can I form various tests comparing the different levels of a categorical variable after anova or regress?
IntroductionOften researchers want to test for differences between levels of a factor (categorical variable) or factors after running an anova or regress command. For instance, with one factor the questions might be
There are often other kinds of tests between levels of a factor that are also of interest. For instance, the questions might be of the form:
Many other interesting questions like these can be asked after an estimation command involving a categorical variable (factor). The test command is one tool to use in answering these questions. There are several variations of the syntax for test depending on if you wish to test coefficients, expressions, terms (after anova), or to test several coefficients at the same time. One form of the test command that we will use is test [exp = exp] [, accumulate notest ] Details can be found in the Reference Manual ([R] test and in the case of anova the sections in [R] anova postestimation that discuss testing). One categorical factorHow you parameterize your model makes a difference in what you do to perform your tests. We will examine a couple of ways of parameterizing a simple one-way ANOVA model. We will use the following three approaches:
With each of these approaches, we will show how to use the test command to obtain tests of interest. In particular, we will
Here is a 20-observation dataset with two variables, the outcome y, and a categorical variable x with four levels. . table x, c(mean y)
----------+-----------
x | mean(y)
----------+-----------
1 | 4.4
2 | 3.4
3 | 5.6
4 | 6.4
----------+-----------
For regress we need to create the indicator (sometimes called “dummy”) variables corresponding to our variable x. We can use xi to create the indicator variables (xi automatically omits one level). Instead, we will use the generate() option of tabulate to produce the indicator variables since it does not automatically omit a level. . quietly tabulate x, gen(cat) . list, noobs +-----------------------------------+ | x y cat1 cat2 cat3 cat4 | |-----------------------------------| | 1 7 1 0 0 0 | | 1 5 1 0 0 0 | | 1 3 1 0 0 0 | | 1 4 1 0 0 0 | | 1 3 1 0 0 0 | |-----------------------------------| | 2 5 0 1 0 0 | | 2 3 0 1 0 0 | | 2 5 0 1 0 0 | | 2 3 0 1 0 0 | | 2 1 0 1 0 0 | |-----------------------------------| | 3 6 0 0 1 0 | | 3 8 0 0 1 0 | | 3 6 0 0 1 0 | | 3 4 0 0 1 0 | | 3 4 0 0 1 0 | |-----------------------------------| | 4 5 0 0 0 1 | | 4 8 0 0 0 1 | | 4 6 0 0 0 1 | | 4 8 0 0 0 1 | | 4 5 0 0 0 1 | +-----------------------------------+ Regression excluding the interceptHere is the regression excluding the intercept. All four levels of x are included in the regression model (possible because we omit the constant). . regress y cat*, noconstant
Source | SS df MS Number of obs = 20
---------+------------------------------ F( 4, 16) = 48.24
Model | 516.2 4 129.05 Prob > F = 0.0000
Residual | 42.8 16 2.675 R-squared = 0.9234
---------+------------------------------ Adj R-squared = 0.9043
Total | 559 20 27.95 Root MSE = 1.6355
------------------------------------------------------------------------------
y | Coef. Std. Err. t P>|t| [95% Conf. Interval]
---------+--------------------------------------------------------------------
cat1 | 4.4 .7314369 6.02 0.000 2.849423 5.950577
cat2 | 3.4 .7314369 4.65 0.000 1.849423 4.950577
cat3 | 5.6 .7314369 7.66 0.000 4.049423 7.150577
cat4 | 6.4 .7314369 8.75 0.000 4.849423 7.950577
------------------------------------------------------------------------------
The coefficients agree with the means reported by table. These coefficients are easily interpreted and easily tested. Here are the three tests after this regression:
Any of several other strange and/or wonderful tests could be performed. If you wish to test a nonlinear expression, you will want to look at testnl (see [R] testnl). If you want to jointly test two or more of these single degree-of-freedom tests, you can use the accumulate option of test. Another useful command is lincom (see [R] lincom). It can also be used to test many of the same hypotheses as the test command and has the benefit of providing not only the test result but also the estimate of the linear combination and the standard error of the estimate along with confidence intervals. Just as an example, here is the third test done using lincom. . lincom 5*cat1 + 4*cat2 - 3*cat3 - 2*cat4
( 1) 5 cat1 + 4 cat2 - 3 cat3 - 2 cat4 = 0.0
--------------------------------------------------------------------------
y | Coef. Std. Err. t P>|t| [95% Conf. Interval]
---------+----------------------------------------------------------------
(1) | 6 5.374942 1.12 0.281 -5.394368 17.39437
--------------------------------------------------------------------------
Regression with the interceptNow, what about this same problem, but after a regression that includes the constant? If you run regress y cat* one of the category variables will be dropped due to collinearity. We can force regress to drop the first category (for example) by leaving that indicator variable out of the model. . regress y cat2-cat4
Source | SS df MS Number of obs = 20
---------+------------------------------ F( 3, 16) = 3.26
Model | 26.15 3 8.71666667 Prob > F = 0.0492
Residual | 42.8 16 2.675 R-squared = 0.3793
---------+------------------------------ Adj R-squared = 0.2629
Total | 68.95 19 3.62894737 Root MSE = 1.6355
------------------------------------------------------------------------------
y | Coef. Std. Err. t P>|t| [95% Conf. Interval]
---------+--------------------------------------------------------------------
cat2 | -1 1.034408 -0.97 0.348 -3.192847 1.192847
cat3 | 1.2 1.034408 1.16 0.263 -.9928471 3.392847
cat4 | 2 1.034408 1.93 0.071 -.1928471 4.192847
_cons | 4.4 .7314369 6.02 0.000 2.849423 5.950577
------------------------------------------------------------------------------
If you compare the coefficients from this regression to the output from the regression without a constant (and to the table showing the mean for each category), you see that the mean of the dropped category (one) corresponds to the coefficient for the constant, and if you add the coefficient for the constant to the other category coefficients, you get the mean for those categories. The coefficients for the included levels are relative to the dropped level. This result provides a clue on how to obtain our three example tests of interest when the constant is in the model. It is a matter of doing some simple algebra to take a test from the first regression model and produce an equivalent test in this regression model. For reference, here is the correspondence between the two regression models:
After our first regression (without a constant), the first test was test cat1 = cat2. After this latest regression (with a constant), the same test is test _cons = _cons + cat2. This test can be simplified to test cat2. The equivalent second and third tests can be similarly determined. Here are the three tests after regress with the constant included:
When you have the constant in the model (and in more complicated designs), it is important to understand how to interpret the coefficients from the regression model so that you can form the correct tests. The formation of the test above is not very intuitive, but becomes clearer after doing some algebra starting with your understanding of the meaning of each coefficient and how they relate to the quantities you wish to test. ANOVAThe anova command is a natural choice for analyzing this same dataset. . anova y x
Number of obs = 20 R-squared = 0.3793
Root MSE = 1.63554 Adj R-squared = 0.2629
Source | Partial SS df MS F Prob > F
-----------+----------------------------------------------------
Model | 26.15 3 8.71666667 3.26 0.0492
|
x | 26.15 3 8.71666667 3.26 0.0492
|
Residual | 42.8 16 2.675
-----------+----------------------------------------------------
Total | 68.95 19 3.62894737
We will examine obtaining individual degrees-of-freedom tests shortly. First, we will take a look at the underlying regression for this anova. You can get it in several ways. You can simply type regress after an anova, or you can type anova, regress (or you could have specified the regress option when originally running the anova). . anova, regress
Source | SS df MS Number of obs = 20
---------+------------------------------ F( 3, 16) = 3.26
Model | 26.15 3 8.71666667 Prob > F = 0.0492
Residual | 42.8 16 2.675 R-squared = 0.3793
---------+------------------------------ Adj R-squared = 0.2629
Total | 68.95 19 3.62894737 Root MSE = 1.6355
------------------------------------------------------------------------------
y Coef. Std. Err. t P>|t| [95% Conf. Interval]
------------------------------------------------------------------------------
_cons 6.4 .7314369 8.75 0.000 4.849423 7.950577
x
1 -2 1.034408 -1.93 0.071 -4.192847 .1928471
2 -3 1.034408 -2.90 0.010 -5.192847 -.8071529
3 -.8 1.034408 -0.77 0.451 -2.992847 1.392847
4 (dropped)
------------------------------------------------------------------------------
From the underlying regression table, anova dropped the fourth level of x (the noconstant option of anova could have been used to exclude the constant if desired). Here, the mean of the dropped level (four) is equal to the coefficient for the constant. The mean of the other three levels is equal to the coefficient for that level plus the coefficient for the constant. After regress, you could say something like test cat1 = cat2, because there is a one-to-one correspondence between variable names and coefficients. After anova, you must say something like test _b[x[1]] = _b[x[2]] (that is, include terms inside _b[] or the synonym _coef[] and indicate the level of the term). With anova a variable corresponds to more than one coefficient. (After regress, we could have also used the _b[] notation, but most people do not.) The test() option of test after anova provides a useful shortcut for specifying these kinds of tests. We will explore that later. Another helpful piece of information is that by definition the coefficient for a dropped level is zero. The following table illustrates how the coefficients from the anova above are related to the mean for each level of x.
The test commands for obtaining our three example tests after the latest anova are shown below:
In the first two tests, the _b[_cons] cancels out and does not need to be included in the test statement at all. In fact, the first two test commands are probably what you would have guessed them to be without ever having looked closely at the underlying regression model. For many of the tests that people typically perform, this will be true. The third (and admittedly stranger) test is different. The constant does not cancel out. If you were to execute your first guess as to the proper test command, you would not have obtained the test you desired. Here you need to look at how the underlying coefficients relate to the quantities of interest to you. In this third test, we allowed Stata’s test command to do the algebra for us. We simply entered the appropriate linear combination of coefficients for each cell mean used in the linear combination being tested. The lincom command also works after anova. For instance, here is the third test using lincom. . lincom 5*(_b[x[1]]+_b[_cons]) + 4*(_b[x[2]]+_b[_cons])
- 3*(_b[x[3]]+_b[_cons]) - 2*(_b[_cons])
( 1) 4 _cons + 5 x[1] + 4 x[2] - 3 x[3] = 0
--------------------------------------------------------------------------
y | Coef. Std. Err. t P>|t| [95% Conf. Interval]
---------+----------------------------------------------------------------
(1) | 6 5.374942 1.12 0.281 -5.394368 17.39437
--------------------------------------------------------------------------
The test() option of test after anova provides a convenient shorthand for specifying these kinds of tests using a matrix to denote the equation. test, showorder lists the order of the terms, and hence, the order of columns for the matrix. . test, showorder
Order of columns in the design matrix
1: _cons
2: (x==1)
3: (x==2)
4: (x==3)
5: (x==4)
Here are the same three tests using the test() option.
Two categorical factorsWhen there are two (or more) categorical factors in our model, we again may want to test various single degrees-of-freedom hypotheses that compare various levels of the two (or more) factors in the model. As with the one-way ANOVA model, how you parameterize your two-way (or higher) model affects how you go about performing individual tests. To demonstrate how to obtain single degrees-of-freedom tests after a two-way ANOVA, we will use the following 24-observation dataset where the variables a and b are categorical variables with 4 and 3 levels, respectively, and there is a response variable, y. . list, noobs sep(6) +------------+ | a b y | |------------| | 1 1 26 | | 1 1 30 | | 1 2 54 | | 1 2 50 | | 1 3 34 | | 1 3 46 | |------------| | 2 1 16 | | 2 1 20 | | 2 2 36 | | 2 2 24 | | 2 3 50 | | 2 3 34 | |------------| | 3 1 48 | | 3 1 28 | | 3 2 28 | | 3 2 28 | | 3 3 50 | | 3 3 46 | |------------| | 4 1 50 | | 4 1 46 | | 4 2 48 | | 4 2 44 | | 4 3 48 | | 4 3 28 | +------------+ The following table shows the mean of y for each cell of a by b as well as the means for each level of a and b (the column and row titled “Total”): . table a b, c(mean y) row col
----------+---------------------------
| b
a | 1 2 3 Total
----------+---------------------------
1 | 28 52 40 40
2 | 18 30 42 30
3 | 38 28 48 38
4 | 48 46 38 44
|
Total | 33 39 42 38
----------+---------------------------
This dataset is balanced (two observations per cell). If you are dealing with unbalanced data (including the case where you have missing cells) you will want to also read the Technical Notes in the section titled Two-way analysis of variance in the [R] anova manual entry. The standard way of performing an ANOVA on this dataset is with . anova y a b a*b
Number of obs = 24 R-squared = 0.7606
Root MSE = 7.74597 Adj R-squared = 0.5412
Source | Partial SS df MS F Prob > F
-----------+----------------------------------------------------
Model | 2288 11 208 3.47 0.0214
|
a | 624 3 208 3.47 0.0509
b | 336 2 168 2.80 0.1005
a*b | 1328 6 221.333333 3.69 0.0259
|
Residual | 720 12 60
-----------+----------------------------------------------------
Total | 3008 23 130.782609
This is the overparameterized two-way ANOVA model (which we will discuss in more detail later). When it comes to single degrees-of-freedom tests, some people prefer to use a different parameterization—the cell means model. Cell means ANOVA modelIn the cell means ANOVA model, we first create one categorical variable that corresponds to the cells in the two-way table (or higher-order table if more than two categorical variables are involved). The egen group() function is useful for creating the single categorical variable. . egen c = group(a b)
. table a b, c(mean c)
----------+-----------------
| b
a | 1 2 3
----------+-----------------
1 | 1 2 3
2 | 4 5 6
3 | 7 8 9
4 | 10 11 12
----------+-----------------
The table above reminds us how the c variable relates to the original a and b variables. For instance, when c is 8, it means that a is 3 and b is 2. (If a and b had been reversed in the egen group() option, then the table above would show a different relationship.) The cell means ANOVA model is then obtained by using the noconstant option of anova and the newly created c variable in place of a and b. (You can also use the regress command to perform the cell means ANOVA model. You create the full set of indicator variables corresponding to c and then use these along with the noconstant option of regress. Here, we will instead concentrate on using the anova command.) . anova y c, nocons
Number of obs = 24 R-squared = 0.9809
Root MSE = 7.74597 Adj R-squared = 0.9618
Source | Partial SS df MS F Prob > F
-----------+----------------------------------------------------
Model | 36944 12 3078.66667 51.31 0.0000
|
c | 36944 12 3078.66667 51.31 0.0000
|
Residual | 720 12 60
-----------+----------------------------------------------------
Total | 37664 24 1569.33333
. anova, regress
Source | SS df MS Number of obs = 24
---------+------------------------------ F( 12, 12) = 51.31
Model | 36944 12 3078.66667 Prob > F = 0.0000
Residual | 720 12 60 R-squared = 0.9809
---------+------------------------------ Adj R-squared = 0.9618
Total | 37664 24 1569.33333 Root MSE = 7.746
------------------------------------------------------------------------------
y Coef. Std. Err. t P>|t| [95% Conf. Interval]
------------------------------------------------------------------------------
c
1 28 5.477226 5.11 0.000 16.06615 39.93385
2 52 5.477226 9.49 0.000 40.06615 63.93385
3 40 5.477226 7.30 0.000 28.06615 51.93385
4 18 5.477226 3.29 0.007 6.066151 29.93385
5 30 5.477226 5.48 0.000 18.06615 41.93385
6 42 5.477226 7.67 0.000 30.06615 53.93385
7 38 5.477226 6.94 0.000 26.06615 49.93385
8 28 5.477226 5.11 0.000 16.06615 39.93385
9 48 5.477226 8.76 0.000 36.06615 59.93385
10 48 5.477226 8.76 0.000 36.06615 59.93385
11 46 5.477226 8.40 0.000 34.06615 57.93385
12 38 5.477226 6.94 0.000 26.06615 49.93385
------------------------------------------------------------------------------
Compare the 12 coefficients for c in the table above with the table of means presented earlier. The coefficients from the cell means ANOVA model are the cell means from the two-way table. This correspondence makes creating meaningful test statements easy. You can recreate an F test from the overparameterized ANOVA model by using appropriate combinations of single degree-of-freedom tests after the cell means ANOVA model. For example, the test for the term a with 3 degrees of freedom can be obtained by accumulating 3 single degree-of-freedom tests. Below I combine the tests of level 1 versus 2, level 1 versus 3, and level 1 versus 4 of the a variable. (Remember that c 1, 2, and 3 correspond to level 1 of a; c 4, 5, and 6 corresponds to level 2 of a; and so on.) . mat amat = (1,1,1,-1,-1,-1,0,0,0,0,0,0 \ ///
1,1,1,0,0,0,-1,-1,-1,0,0,0 \ ///
1,1,1,0,0,0,0,0,0,-1,-1,-1)
. test, test(amat)
( 1) c[1] + c[2] + c[3] - c[4] - c[5] - c[6] = 0
( 2) c[1] + c[2] + c[3] - c[7] - c[8] - c[9] = 0
( 3) c[1] + c[2] + c[3] - c[10] - c[11] - c[12] = 0
F( 3, 12) = 3.47
Prob > F = 0.0509
This F of 3.47 agrees with the F test for the term a in the overparameterized ANOVA presented earlier. Of course, it is easier to obtain the test of a term like a by running the overparameterized model. I just wanted to show that you could also obtain the result with a little work starting from the cell means model. For the sake of completeness, here is the test command to produce the F test for the b term. . mat bmat = (1,-1,0,1,-1,0,1,-1,0,1,-1,0 \ ///
1,0,-1,1,0,-1,1,0,-1,1,0,-1)
. test, test(bmat)
( 1) c[1] - c[2] + c[4] - c[5] + c[7] - c[8] + c[10] - c[11] = 0
( 2) c[1] - c[3] + c[4] - c[6] + c[7] - c[9] + c[10] - c[12] = 0
F( 2, 12) = 2.80
Prob > F = 0.1005
Here is the test command to produce the F test for the a by b interaction term. . forvalues i = 1/3 {
2. forvalues j = 1/2 {
3. mat abmat = nullmat(abmat) \ vecdiag(amat[`i',1...]'*bmat[`j',1...])
4. }
5. }
. mat list abmat
abmat[6,12]
c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 c11 c12
r1 1 -1 0 -1 1 0 0 0 0 0 0 0
r1 1 0 -1 -1 0 1 0 0 0 0 0 0
r1 1 -1 0 0 0 0 -1 1 0 0 0 0
r1 1 0 -1 0 0 0 -1 0 1 0 0 0
r1 1 -1 0 0 0 0 0 0 0 -1 1 0
r1 1 0 -1 0 0 0 0 0 0 -1 0 1
. test, test(abmat)
( 1) c[1] - c[2] - c[4] + c[5] = 0
( 2) c[1] - c[3] - c[4] + c[6] = 0
( 3) c[1] - c[2] - c[7] + c[8] = 0
( 4) c[1] - c[3] - c[7] + c[9] = 0
( 5) c[1] - c[2] - c[10] + c[11] = 0
( 6) c[1] - c[3] - c[10] + c[12] = 0
F( 6, 12) = 3.69
Prob > F = 0.0259
There are actually many different ways that I could have obtained the overall F tests for the a, b, and a by b terms. Now that we have demonstrated that you can reproduce the results from the overparameterized model with an appropriate series of test statements after a cell means model, let us now look at a few different single degrees-of-freedom tests. (We will later see how to obtain these same single degrees-of-freedom tests after the overparameterized ANOVA.) You will want to look back at the table showing how c relates to a and b to see how these tests were constructed.
Those same four tests using the test() option are
Constructing various single degrees-of-freedom tests after a cell means ANOVA model is relatively easy. You pick the appropriate linear combination of the coefficients based on how the single categorical variable (c in our example) relates to the original categorical variables (a and b in our example) and based on the hypothesis of interest. Overparameterized ANOVA modelMost people are used to the results presented by the overparameterized ANOVA model. As we saw when we discussed the cell means ANOVA model, the F tests for terms in the ANOVA model are obtained directly from the overparameterized model ANOVA table. Compare this with computing an F test for a term after the cell means ANOVA model. However, when it comes to obtaining single degrees-of-freedom tests, most people find the cell means model approach to be the easiest. Here again is the overparameterized ANOVA model for our example data. Also, I use the regress command to replay the ANOVA as a regression table (I could have also said anova, regress to see this display). When you see the various levels reported as “dropped”, you begin to understand why it is called the overparameterized ANOVA model. . anova y a b a*b
Number of obs = 24 R-squared = 0.7606
Root MSE = 7.74597 Adj R-squared = 0.5412
Source | Partial SS df MS F Prob > F
-----------+----------------------------------------------------
Model | 2288 11 208 3.47 0.0214
|
a | 624 3 208 3.47 0.0509
b | 336 2 168 2.80 0.1005
a*b | 1328 6 221.333333 3.69 0.0259
|
Residual | 720 12 60
-----------+----------------------------------------------------
Total | 3008 23 130.782609
. regress, noheader
------------------------------------------------------------------------------
y Coef. Std. Err. t P>|t| [95% Conf. Interval]
------------------------------------------------------------------------------
_cons 38 5.477226 6.94 0.000 26.06615 49.93385
a
1 2 7.745967 0.26 0.801 -14.87701 18.87701
2 4 7.745967 0.52 0.615 -12.87701 20.87701
3 10 7.745967 1.29 0.221 -6.877012 26.87701
4 (dropped)
b
1 10 7.745967 1.29 0.221 -6.877012 26.87701
2 8 7.745967 1.03 0.322 -8.877012 24.87701
3 (dropped)
a*b
1 1 -22 10.95445 -2.01 0.068 -45.8677 1.867699
1 2 4 10.95445 0.37 0.721 -19.8677 27.8677
1 3 (dropped)
2 1 -34 10.95445 -3.10 0.009 -57.8677 -10.1323
2 2 -20 10.95445 -1.83 0.093 -43.8677 3.867699
2 3 (dropped)
3 1 -20 10.95445 -1.83 0.093 -43.8677 3.867699
3 2 -28 10.95445 -2.56 0.025 -51.8677 -4.132301
3 3 (dropped)
4 1 (dropped)
4 2 (dropped)
4 3 (dropped)
------------------------------------------------------------------------------
Now the important question is how the coefficients in this model relate to the cell means. To refresh your memory, here is the table of cell means (and marginal means). . table a b, c(mean y) row col
----------+---------------------------
| b
a | 1 2 3 Total
----------+---------------------------
1 | 28 52 40 40
2 | 18 30 42 30
3 | 38 28 48 38
4 | 48 46 38 44
|
Total | 33 39 42 38
----------+---------------------------
The cell mean for level i of a and level j of b is equal to the coefficient for the constant plus the coefficient for a at level i, plus the coefficient for b at level j, plus the coefficient for a and b at i and j. When a coefficient is dropped in the regression table, the corresponding coefficient is zero. The table below shows the relationship.
The simplifications shown at the far right of the table are due to the coefficients that are dropped from the overparameterized model being zero. The marginal means can easily be built up by averaging appropriate cell means together. We can obtain the same four single degrees-of-freedom tests as were obtained with the cell means ANOVA model by examining the relationship (shown in the table above) between the coefficients of the overparameterized model and the quantities of real interest—the cell means. We could simply plug in all the coefficients for each cell involved in the test and let Stata’s test command do the algebra, or we can do the simplifying ourselves. For the same four tests that were performed for the cell means model, I will show the results when you plug everything into test (based on the simplification in the far right column of the table above) and let Stata’s test command do the algebra. (It also would work if I plugged the unsimplified cell mean expressions into test.)
You can compare these results with those obtained after the cell means ANOVA model to see that they are the same. The test() option of test may also be used. First we check the order of the columns. . test, showorder
Order of columns in the design matrix
1: _cons
2: (a==1)
3: (a==2)
4: (a==3)
5: (a==4)
6: (b==1)
7: (b==2)
8: (b==3)
9: (a==1)*(b==1)
10: (a==1)*(b==2)
11: (a==1)*(b==3)
12: (a==2)*(b==1)
13: (a==2)*(b==2)
14: (a==2)*(b==3)
15: (a==3)*(b==1)
16: (a==3)*(b==2)
17: (a==3)*(b==3)
18: (a==4)*(b==1)
19: (a==4)*(b==2)
20: (a==4)*(b==3)
Now here are the same four tests.
How did we come up with matrices x1–x4? Let me illustrate with x4.
The column titled “Sum” is the sum over the previous five columns and provides the row elements of the matrix x4. Somtimes it is easiest to specify the equation, and other times it is easier to specify the corresponding matrix in the test() option. Either method works. |
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