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This page was adapted from a FAQ at the Stata Corp. FAQ page. We thank Stata for their permission to adapt and distribute this page via our web site.
Often 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:
There are many other interesting questions like these that are possible to ask 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. The 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 that discuss testing).
The way in which 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
Here is the regression excluding the intercept. Note that 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 | 516.20 4 129.05 Prob > F = 0.0000
Residual | 42.80 16 2.675 R-squared = 0.9234
---------+------------------------------ Adj R-squared = 0.9043
Total | 559.00 20 27.95 Root MSE = 1.6355
------------------------------------------------------------------------------
y | Coef. Std. Err. t P>|t| [95% Conf. Interval]
---------+--------------------------------------------------------------------
cat1 | 4.4 .7314369 6.016 0.000 2.849423 5.950577
cat2 | 3.4 .7314369 4.648 0.000 1.849423 4.950577
cat3 | 5.6 .7314369 7.656 0.000 4.049423 7.150577
cat4 | 6.4 .7314369 8.750 0.000 4.849423 7.950577
------------------------------------------------------------------------------
Notice how 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:
test cat1 = cat2
( 1) cat1 - cat2 = 0.0
F( 1, 16) = 0.93
Prob > F = 0.3481test 0.5*cat1 + 0.5*cat2 = cat3( 1) .5 cat1 + .5 cat2 - cat3 = 0.0 F( 1, 16) = 3.60 Prob > F = 0.0759
test 5*cat1 + 4*cat2 - 3*cat3 = 2*cat4( 1) 5.0 cat1 + 4.0 cat2 - 3.0 cat3 - 2.0 cat4 = 0.0 F( 1, 16) = 1.25 Prob > F = 0.2808
Any of a number of 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 the estimate of the linear combination and the standard error of the estimate along with confidence intervals. However, lincom will not allow additive constants and can not be used after anova. Just as an example, here is that third test done using lincom.
lincom 5*cat1 + 4*cat2 - 3*cat3 - 2*cat4
( 1) 5.0 cat1 + 4.0 cat2 - 3.0 cat3 - 2.0 cat4 = 0.0
------------------------------------------------------------------------------
y | Coef. Std. Err. t P>|t| [95% Conf. Interval]
---------+--------------------------------------------------------------------
(1) | 6 5.374942 1.116 0.281 -5.394368 17.39437
------------------------------------------------------------------------------
Now, 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.
Source | SS df MS Number of obs = 20
---------+------------------------------ F( 3, 16) = 3.26
Model | 26.15 3 8.71666667 Prob > F = 0.0492
Residual | 42.80 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 -0.967 0.348 -3.192847 1.192847
cat3 | 1.2 1.034408 1.160 0.263 -.9928471 3.392847
cat4 | 2 1.034408 1.933 0.071 -.1928471 4.192847
_cons | 4.4 .7314369 6.016 0.000 2.849423 5.950577
------------------------------------------------------------------------------
regress y cat2-cat4
Notice that 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 that if you add the coefficient for the constant to the other category coefficients you get the mean for those categories. In other words, the coefficients for the included levels are relative to the dropped level.
This provides the 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:
| No constant | With constant |
|---|---|
| cat1 | _cons |
| cat2 | _cons + cat2 |
| cat3 | _cons + cat3 |
| cat4 | _cons + cat4 |
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 determined in a similar fashion. Here are the three tests after regress with the constant included:
test cat2( 1) cat2 = 0.0 F( 1, 16) = 0.93 Prob > F = 0.3481
test 0.5*cat2 = cat3( 1) .5 cat2 - cat3 = 0.0 F( 1, 16) = 3.60 Prob > F = 0.0759
test 4*_cons + 4*cat2 - 3*cat3 = 2*cat4( 1) 4.0 cat2 - 3.0 cat3 - 2.0 cat4 + 4.0 _cons = 0.0 F( 1, 16) = 1.25 Prob > F = 0.2808
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.
The anova command is a natural choice for analyzing this same data.
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 | 42.80 16 2.675
-----------+----------------------------------------------------
Total | 68.95 19 3.62894737
We will examine obtaining individual degree-of-freedom tests in just a moment. 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 | 42.80 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 8.750 0.000 4.849423 7.950577
x
1 -2 1.034408 -1.933 0.071 -4.192847 .1928471
2 -3 1.034408 -2.900 0.010 -5.192847 -.8071529
3 -.8 1.034408 -0.773 0.451 -2.992847 1.392847
4 (dropped)
------------------------------------------------------------------------------
From the underlying regression table, notice that anova dropped the fourth level of x (the noconstant option of anova could have been used to exclude the constant if desired). In our case, 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 of _b[] or the synonym _coef[] and indicate the level of the term). This is because 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.)
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.
| Mean of x | anova coefficients |
|---|---|
| level 1 | _b[x[1]]+_b[_cons] |
| level 2 | _b[x[2]]+_b[_cons] |
| level 3 | _b[x[3]]+_b[_cons] |
| level 4 | _b[_cons] |
The test commands for obtaining our three example tests after the latest anova are shown below:
test _b[x[1]] = _b[x[2]]( 1) x[1] - x[2] = 0.0 F( 1, 16) = 0.93 Prob > F = 0.3481
test 0.5*_b[x[1]] + 0.5*_b[x[2]] = _b[x[3]]( 1) .5 x[1] + .5 x[2] - x[3] = 0.0 F( 1, 16) = 3.60 Prob > F = 0.0759
test 5*(_b[x[1]]+_b[_cons]) + 4*(_b[x[2]]+_b[_cons]) - 3*(_b[x[3]]+
_b[_cons]) = 2*(_b[_cons])
( 1) 4.0 _cons + 5.0 x[1] + 4.0 x[2] - 3.0 x[3] = 0.0
F( 1, 16) = 1.25
Prob > F = 0.2808
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. And 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. In this case you need to look at how the underlying coefficients relate to the quantities of interest to you. Notice that 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.
When there are two (or more) categorical factors in our model we again may want to test various single degree-of-freedom hypotheses that compare various levels of the two (or more) factors in the model. As with the one-way ANOVA model, the way in which you parameterize your two-way (or higher) model affects how you go about performing individual tests.
To demonstrate how to obtain single degree-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
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 data 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 | 2288.00 11 208.00 3.47 0.0214
|
a | 624.00 3 208.00 3.47 0.0509
b | 336.00 2 168.00 2.80 0.1005
a*b | 1328.00 6 221.333333 3.69 0.0259
|
Residual | 720.00 12 60.00
-----------+----------------------------------------------------
Total | 3008.00 23 130.782609
This is the overparameterized two-way ANOVA model (which we will discuss in
more detail later). When it comes to single degree-of-freedom tests, some
people prefer to use a different parameterization – the cell means model.
In 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.
(It is, of course, also possible to 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 | 36944.00 12 3078.66667 51.31 0.0000
|
c | 36944.00 12 3078.66667 51.31 0.0000
|
Residual | 720.00 12 60.00
-----------+----------------------------------------------------
Total | 37664.00 24 1569.33333
anova, regress
Source | SS df MS Number of obs = 24
---------+------------------------------ F( 12, 12) = 51.31
Model | 36944.00 12 3078.66667 Prob > F = 0.0000
Residual | 720.00 12 60.00 R-squared = 0.9809
---------+------------------------------ Adj R-squared = 0.9618
Total | 37664.00 24 1569.33333 Root MSE = 7.746
------------------------------------------------------------------------------
y Coef. Std. Err. t P>|t| [95% Conf. Interval]
------------------------------------------------------------------------------
c
1 28 5.477226 5.112 0.000 16.06615 39.93385
2 52 5.477226 9.494 0.000 40.06615 63.93385
3 40 5.477226 7.303 0.000 28.06615 51.93385
4 18 5.477226 3.286 0.007 6.066151 29.93385
5 30 5.477226 5.477 0.000 18.06615 41.93385
6 42 5.477226 7.668 0.000 30.06615 53.93385
7 38 5.477226 6.938 0.000 26.06615 49.93385
8 28 5.477226 5.112 0.000 16.06615 39.93385
9 48 5.477226 8.764 0.000 36.06615 59.93385
10 48 5.477226 8.764 0.000 36.06615 59.93385
11 46 5.477226 8.398 0.000 34.06615 57.93385
12 38 5.477226 6.938 0.000 26.06615 49.93385
------------------------------------------------------------------------------
Compare the 12 coefficients for c in the table above to 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 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 correspond to level 2 of a, and so on.)
test _b[c[1]] + _b[c[2]] + _b[c[3]] = _b[c[4]] + _b[c[5]] + _b[c[6]]
( 1) c[1] + c[2] + c[3] - c[4] - c[5] - c[6] = 0.0
F( 1, 12) = 5.00
Prob > F = 0.0451
test _b[c[1]] + _b[c[2]] + _b[c[3]] = _b[c[7]] + _b[c[8]] + _b[c[9]], accum
( 1) c[1] + c[2] + c[3] - c[4] - c[5] - c[6] = 0.0
( 2) c[1] + c[2] + c[3] - c[7] - c[8] - c[9] = 0.0
F( 2, 12) = 2.80
Prob > F = 0.1005
test _b[c[1]] + _b[c[2]] + _b[c[3]] = _b[c[10]] + _b[c[11]] + _b[c[12]], accum
( 1) c[1] + c[2] + c[3] - c[4] - c[5] - c[6] = 0.0
( 2) c[1] + c[2] + c[3] - c[7] - c[8] - c[9] = 0.0
( 3) c[1] + c[2] + c[3] - c[10] - c[11] - c[12] = 0.0
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 over-parameterized ANOVA presented earlier. Of course, it is easier to obtain the test of a term like a by running the over-parameterized model. I just wanted to show that you could also obtain the result with a little work starting from the cell means model.
For completeness sake, here is a set of test commands to produce the F-test for the b term.
Here is a set of test commands to produce the F-test for the a by b interaction term.test _b[c[1]]+_b[c[4]]+_b[c[7]]+_b[c[10]]=_b[c[2]]+_b[c[5]]+_b[c[8]]+_b[c[11]] test _b[c[1]]+_b[c[4]]+_b[c[7]]+_b[c[10]]=_b[c[3]]+_b[c[6]]+_b[c[9]]+_b[c[12]], accum
test _b[c[1]] + _b[c[5]] = _b[c[2]] + _b[c[4]] test _b[c[1]] + _b[c[6]] = _b[c[3]] + _b[c[4]], accum test _b[c[1]] + _b[c[8]] = _b[c[2]] + _b[c[7]], accum test _b[c[1]] + _b[c[9]] = _b[c[3]] + _b[c[7]], accum test _b[c[1]] + _b[c[11]] = _b[c[2]] + _b[c[10]], accum test _b[c[1]] + _b[c[12]] = _b[c[3]] + _b[c[10]], accum
There are actually many different ways I could have combined various single degree-of-freedom tests to obtain 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 over-parameterized model with an appropriate series of test statements after a cell means model, let us now look at a few different single degree-of-freedom tests. (We will later see how to obtain these same single degree-of-freedom tests after the over-parameterized 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.
test _b[c[4]] + _b[c[5]] + _b[c[6]] = _b[c[10]] + _b[c[11]] + _b[c[12]]( 1) c[4] + c[5] + c[6] - c[10] - c[11] - c[12] = 0.0 F( 1, 12) = 9.80 Prob > F = 0.0087
test (_b[c[1]] + _b[c[4]] + _b[c[7]] + _b[c[10]] + _b[c[2]] + _b[c[5]] +
_b[c[8]] + _b[c[11]])/2 = _b[c[3]] + _b[c[6]] + _b[c[9]] + _b[c[12]]
( 1) .5 c[1] + .5 c[2] - c[3] + .5 c[4] + .5 c[5] - c[6] + .5 c[7] + .5 c[8]
- c[9] + .5 c[10] + .5 c[11] - c[12] = 0.0
F( 1, 12) = 3.20
Prob > F = 0.0989test (_b[c[1]] + _b[c[4]] + _b[c[2]] + _b[c[5]])/2 = _b[c[9]] + _b[c[12]]( 1) .5 c[1] + .5 c[2] + .5 c[4] + .5 c[5] - c[9] - c[12] = 0.0 F( 1, 12) = 5.38 Prob > F = 0.0388
test 3*_b[c[1]] - 4*_b[c[8]] + 6*_b[c[12]] = _b[c[5]] - 2*_b[c[6]]( 1) 3.0 c[1] - c[5] + 2.0 c[6] - 4.0 c[8] + 6.0 c[12] = 0.0 F( 1, 12) = 32.58 Prob > F = 0.0001
Constructing various single degree-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.
Most 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 to computing an F-test for a term by accumulating the results of several individual degree-of-freedom tests after the cell means ANOVA model. However, when it comes to obtaining single degree-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. In addition, 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 | 2288.00 11 208.00 3.47 0.0214 | a | 624.00 3 208.00 3.47 0.0509 b | 336.00 2 168.00 2.80 0.1005 a*b | 1328.00 6 221.333333 3.69 0.0259 | Residual | 720.00 12 60.00 -----------+---------------------------------------------------- Total | 3008.00 23 130.782609 regress, noheader------------------------------------------------------------------------------ y Coef. Std. Err. t P>|t| [95% Conf. Interval] ------------------------------------------------------------------------------ _cons 38 5.477226 6.938 0.000 26.06615 49.93385 a 1 2 7.745967 0.258 0.801 -14.87701 18.87701 2 4 7.745967 0.516 0.615 -12.87701 20.87701 3 10 7.745967 1.291 0.221 -6.877011 26.87701 4 (dropped) b 1 10 7.745967 1.291 0.221 -6.877011 26.87701 2 8 7.745967 1.033 0.322 -8.877011 24.87701 3 (dropped) a*b 1 1 -22 10.95445 -2.008 0.068 -45.8677 1.867698 1 2 4 10.95445 0.365 0.721 -19.8677 27.8677 1 3 (dropped) 2 1 -34 10.95445 -3.104 0.009 -57.8677 -10.1323 2 2 -20 10.95445 -1.826 0.093 -43.8677 3.867698 2 3 (dropped) 3 1 -20 10.95445 -1.826 0.093 -43.8677 3.867698 3 2 -28 10.95445 -2.556 0.025 -51.8677 -4.132302 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.
| a | b | cell mean | cell mean (simplified) |
|---|---|---|---|
| a = 1 | b = 1 | _b[_cons]+_b[a[1]]+_b[b[1]]+_b[a[1]*b[1]] | _b[_cons]+_b[a[1]]+_b[b[1]]+_b[a[1]*b[1]] |
| a = 1 | b = 2 | _b[_cons]+_b[a[1]]+_b[b[2]]+_b[a[1]*b[2]] | _b[_cons]+_b[a[1]]+_b[b[2]]+_b[a[1]*b[2]] |
| a = 1 | b = 3 | _b[_cons]+_b[a[1]]+_b[b[3]]+_b[a[1]*b[3]] | _b[_cons]+_b[a[1]] |
| a = 2 | b = 1 | _b[_cons]+_b[a[2]]+_b[b[1]]+_b[a[2]*b[1]] | _b[_cons]+_b[a[2]]+_b[b[1]]+_b[a[2]*b[1]] |
| a = 2 | b = 2 | _b[_cons]+_b[a[2]]+_b[b[2]]+_b[a[2]*b[2]] | _b[_cons]+_b[a[2]]+_b[b[2]]+_b[a[2]*b[2]] |
| a = 2 | b = 3 | _b[_cons]+_b[a[2]]+_b[b[3]]+_b[a[2]*b[3]] | _b[_cons]+_b[a[2]] |
| a = 3 | b = 1 | _b[_cons]+_b[a[3]]+_b[b[1]]+_b[a[3]*b[1]] | _b[_cons]+_b[a[3]]+_b[b[1]]+_b[a[3]*b[1]] |
| a = 3 | b = 2 | _b[_cons]+_b[a[3]]+_b[b[2]]+_b[a[3]*b[2]] | _b[_cons]+_b[a[3]]+_b[b[2]]+_b[a[3]*b[2]] |
| a = 3 | b = 3 | _b[_cons]+_b[a[3]]+_b[b[3]]+_b[a[3]*b[3]] | _b[_cons]+_b[a[3]] |
| a = 4 | b = 1 | _b[_cons]+_b[a[4]]+_b[b[1]]+_b[a[4]*b[1]] | _b[_cons]+_b[b[1]] |
| a = 4 | b = 2 | _b[_cons]+_b[a[4]]+_b[b[2]]+_b[a[4]*b[2]] | _b[_cons]+_b[b[2]] |
| a = 4 | b = 3 | _b[_cons]+_b[a[4]]+_b[b[3]]+_b[a[4]*b[3]] | _b[_cons] |
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 degree-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.)
test (_b[_cons]+_b[a[2]]+_b[b[1]]+_b[a[2]*b[1]]) + (_b[_cons]+_b[a[2]]+
_b[b[2]]+_b[a[2]*b[2]]) + (_b[_cons]+_b[a[2]]) = (_b[_cons]+_b[b[1]]) +
(_b[_cons]+_b[b[2]]) + (_b[_cons])
( 1) 3.0 a[2] + a[2]*b[1] + a[2]*b[2] = 0.0
F( 1, 12) = 9.80
Prob > F = 0.0087 test ((_b[_cons]+_b[a[1]]+_b[b[1]]+_b[a[1]*b[1]]) + (_b[_cons]+_b[a[2]]+
_b[b[1]]+_b[a[2]*b[1]]) + (_b[_cons]+_b[a[3]]+_b[b[1]]+_b[a[3]*b[1]]) +
(_b[_cons]+_b[b[1]]) + (_b[_cons]+_b[a[1]]+_b[b[2]]+_b[a[1]*b[2]]) +
(_b[_cons]+_b[a[2]]+_b[b[2]]+_b[a[2]*b[2]]) + (_b[_cons]+_b[a[3]]+
_b[b[2]]+_b[a[3]*b[2]]) + (_b[_cons]+_b[b[2]]))/2 = (_b[_cons]+_b[a[1]])
+ (_b[_cons]+_b[a[2]]) + (_b[_cons]+_b[a[3]]) + (_b[_cons])
( 1) 2.0 b[1] + 2.0 b[2] + .5 a[1]*b[1] + .5 a[1]*b[2] + .5 a[2]*b[1] +
5 a[2]*b[2] + .5 a[3]*b[1] + .5 a[3]*b[2] = 0.0
F( 1, 12) = 3.20
Prob > F = 0.0989 test ((_b[_cons]+_b[a[1]]+_b[b[1]]+_b[a[1]*b[1]]) + (_b[_cons]+_b[a[2]]+
_b[b[1]]+_b[a[2]*b[1]]) + (_b[_cons]+_b[a[1]]+_b[b[2]]+_b[a[1]*b[2]]) +
(_b[_cons]+_b[a[2]]+_b[b[2]]+_b[a[2]*b[2]]))/2 = (_b[_cons]+_b[a[3]]) +
(_b[_cons])
( 1) a[1] + a[2] - a[3] + b[1] + b[2] + .5 a[1]*b[1] + .5 a[1]*b[2] +
5 a[2]*b[1] + .5 a[2]*b[2] = 0.0
F( 1, 12) = 5.38
Prob > F = 0.0388
test 3*(_b[_cons]+_b[a[1]]+_b[b[1]]+_b[a[1]*b[1]]) - 4*(_b[_cons]+_b[a[3]]+
_b[b[2]]+_b[a[3]*b[2]]) + 6*(_b[_cons]) = (_b[_cons]+_b[a[2]]+_b[b[2]]+
_b[a[2]*b[2]]) - 2*(_b[_cons]+_b[a[2]])
( 1) 6.0 _cons + 3.0 a[1] + a[2] - 4.0 a[3] + 3.0 b[1] - 5.0 b[2] +
3.0 a[1]*b[1] - a[2]*b[2] - 4.0 a[3]*b[2] = 0.0
F( 1, 12) = 32.58
Prob > F = 0.0001
You can compare these results to those obtained after the cell means ANOVA model to see that they are the same.
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