McCullagh and Nelder (1989, Ch. 14.5) describe a mating experiment—conducted by S. Arnold and P. Verell at the University of Chicago, Department of Ecology and Evolution—involving two geographically isolated populations of mountain dusky salamanders. One goal of the experiment was to determine whether barriers to interbreeding have evolved in light of the geographical isolation of the populations. In this case, matings within a population should be more successful than matings between the populations. The experiment conducted in the summer of 1986 involved 40 animals, 20 rough butt (R) and 20 whiteside (W) salamanders, with equal numbers of males and females. The animals were grouped into two sets of R males, two sets of R females, two sets of W males, and two sets of W females, so that each set comprised five salamanders. Each set was mated against one rough butt and one whiteside set, creating eight crossings. Within the pairings of sets, each female was paired to three male animals. The salamander mating data have been used by a number of authors; see, for example, McCullagh and Nelder (1989); Schall (1991); Karim and Zeger (1992); Breslow and Clayton (1993); Wolfinger and O’Connell (1993); Shun (1997).
The following DATA step creates the data set for the analysis.
data salamander; input day fpop$ fnum mpop$ mnum mating @@; datalines; 4 rb 1 rb 1 1 4 rb 2 rb 5 1 4 rb 3 rb 2 1 4 rb 4 rb 4 1 4 rb 5 rb 3 1 4 rb 6 ws 9 1 4 rb 7 ws 8 0 4 rb 8 ws 6 0 4 rb 9 ws 10 0 4 rb 10 ws 7 0 4 ws 1 rb 9 0 4 ws 2 rb 7 0 4 ws 3 rb 8 0 4 ws 4 rb 10 0 4 ws 5 rb 6 0 4 ws 6 ws 5 0 4 ws 7 ws 4 1 4 ws 8 ws 1 1 4 ws 9 ws 3 1 4 ws 10 ws 2 1 8 rb 1 ws 4 1 8 rb 2 ws 5 1 8 rb 3 ws 1 0 8 rb 4 ws 2 1 8 rb 5 ws 3 1 8 rb 6 rb 9 1 8 rb 7 rb 8 0 8 rb 8 rb 6 1 8 rb 9 rb 7 0 8 rb 10 rb 10 0 8 ws 1 ws 9 1 8 ws 2 ws 6 0 8 ws 3 ws 7 0 8 ws 4 ws 10 1 8 ws 5 ws 8 1 8 ws 6 rb 2 0 8 ws 7 rb 1 1 8 ws 8 rb 4 0 8 ws 9 rb 3 1 8 ws 10 rb 5 0 12 rb 1 rb 5 1 12 rb 2 rb 3 1 12 rb 3 rb 1 1 12 rb 4 rb 2 1 12 rb 5 rb 4 1 12 rb 6 ws 10 1 12 rb 7 ws 9 0 12 rb 8 ws 7 0 12 rb 9 ws 8 1 12 rb 10 ws 6 1 12 ws 1 rb 7 1 12 ws 2 rb 9 0 12 ws 3 rb 6 0 12 ws 4 rb 8 1 12 ws 5 rb 10 0 12 ws 6 ws 3 1 12 ws 7 ws 5 1 12 ws 8 ws 2 1 12 ws 9 ws 1 1 12 ws 10 ws 4 0 16 rb 1 ws 1 0 16 rb 2 ws 3 1 16 rb 3 ws 4 1 16 rb 4 ws 5 0 16 rb 5 ws 2 1 16 rb 6 rb 7 0 16 rb 7 rb 9 1 16 rb 8 rb 10 0 16 rb 9 rb 6 1 16 rb 10 rb 8 0 16 ws 1 ws 10 1 16 ws 2 ws 7 1 16 ws 3 ws 9 0 16 ws 4 ws 8 1 16 ws 5 ws 6 0 16 ws 6 rb 4 0 16 ws 7 rb 2 0 16 ws 8 rb 5 0 16 ws 9 rb 1 1 16 ws 10 rb 3 1 20 rb 1 rb 4 1 20 rb 2 rb 1 1 20 rb 3 rb 3 1 20 rb 4 rb 5 1 20 rb 5 rb 2 1 20 rb 6 ws 6 1 20 rb 7 ws 7 0 20 rb 8 ws 10 1 20 rb 9 ws 9 1 20 rb 10 ws 8 1 20 ws 1 rb 10 0 20 ws 2 rb 6 0 20 ws 3 rb 7 0 20 ws 4 rb 9 0 20 ws 5 rb 8 0 20 ws 6 ws 2 0 20 ws 7 ws 1 1 20 ws 8 ws 5 1 20 ws 9 ws 4 1 20 ws 10 ws 3 1 24 rb 1 ws 5 1 24 rb 2 ws 2 1 24 rb 3 ws 3 1 24 rb 4 ws 4 1 24 rb 5 ws 1 1 24 rb 6 rb 8 1 24 rb 7 rb 6 0 24 rb 8 rb 9 1 24 rb 9 rb 10 1 24 rb 10 rb 7 0 24 ws 1 ws 8 1 24 ws 2 ws 10 0 24 ws 3 ws 6 1 24 ws 4 ws 9 1 24 ws 5 ws 7 0 24 ws 6 rb 1 0 24 ws 7 rb 5 1 24 ws 8 rb 3 0 24 ws 9 rb 4 0 24 ws 10 rb 2 0 ;
The first observation, for example, indicates that rough butt female 1 was paired in the laboratory on day 4 of the experiment with rough butt male 1, and the pair mated. On the same day rough butt female 7 was paired with whiteside male 8, but the pairing did not result in mating of the animals.
The model adopted by many authors for these data comprises fixed effects for gender and population, their interaction, and male and female random effects. Specifically, let , , , and denote the mating probabilities between the populations, where the first subscript identifies the female partner of the pair. Then, you model
where and are independent random variables representing female and male random effects (20 each), and denotes the average logit of mating between females of population k and males of population l. The following statements fit this model by pseudo-likelihood:
proc glimmix data=salamander; class fpop fnum mpop mnum; model mating(event='1') = fpop|mpop / dist=binary; random fpop*fnum mpop*mnum; lsmeans fpop*mpop / ilink; run;
The response variable is the two-level variable mating
. Because it is coded as zeros and ones, and because PROC GLIMMIX models by default the probability of the first level according
to the response-level ordering, the EVENT=’1’ option instructs PROC GLIMMIX to model the probability of a successful mating.
The distribution of the mating variable, conditional on the random effects, is binary.
The fpop*fnum
effect in the RANDOM statement creates a random intercept for each female animal. Because fpop
and fnum
are CLASS variables, the effect has 20 levels (10 rb and 10 ws females). Similarly, the mpop*mnum
effect creates the random intercepts for the male animals. Because no TYPE= is specified in the RANDOM statement, the covariance structure defaults to TYPE=VC. The random effects and their levels are independent, and each effect has its own variance component. Because the conditional
distribution of the data, conditioned on the random effects, is binary, no
extra scale parameter () is added.
The LSMEANS statement requests least squares means for the four levels of the fpop*mpop
effect, which are estimates of the cell means in the classification of female and male populations. The ILINK option in the LSMEANS statement requests that the estimated means and standard errors are also reported on the scale of the data. This yields estimates
of the four mating probabilities, , , , and .
The “Model Information” table displays general information about the model being fit (Output 43.2.1).
Output 43.2.1: Analysis of Mating Experiment with Crossed Random Effects
Model Information | |
---|---|
Data Set | WORK.SALAMANDER |
Response Variable | mating |
Response Distribution | Binary |
Link Function | Logit |
Variance Function | Default |
Variance Matrix | Not blocked |
Estimation Technique | Residual PL |
Degrees of Freedom Method | Containment |
The response variable mating
follows a binary distribution (conditional on the random effects). Hence, the mean of the data is an event probability, , and the logit of this probability is linearly related to the linear predictor of the model. The variance function is the
default function that is implied by the distribution, . The variance matrix is not blocked, because the GLIMMIX procedure does not process the data by subjects (see the section
Processing by Subjects for details). The estimation technique is the default method for GLMMs, residual pseudo-likelihood (METHOD=RSPL), and degrees of freedom for tests and confidence intervals are determined by the containment method.
The “Class Level Information” table in Output 43.2.2 lists the levels of the variables listed in the CLASS statement, as well as the order of the levels.
Output 43.2.2: Class Level Information and Number of Observations
Class Level Information | ||
---|---|---|
Class | Levels | Values |
fpop | 2 | rb ws |
fnum | 10 | 1 2 3 4 5 6 7 8 9 10 |
mpop | 2 | rb ws |
mnum | 10 | 1 2 3 4 5 6 7 8 9 10 |
Number of Observations Read | 120 |
---|---|
Number of Observations Used | 120 |
Note that there are two female populations and two male populations; also, the variables fnum
and mnum
have 10 levels each. As a consequence, the effects fpop*fnum
and mpop*mnum
identify the 20 females and males, respectively. The effect fpop*mpop
identifies the four mating types.
The “Response Profile Table,” which is displayed for binary or multinomial data, lists the levels of the response variable and their order (Output 43.2.3). With binary data, the table also provides information about which level of the response variable defines the event. Because of the EVENT=’1’ response variable option in the MODEL statement, the probability being modeled is that of the higher-ordered value.
Output 43.2.3: Response Profiles
Response Profile | ||
---|---|---|
Ordered Value |
mating | Total Frequency |
1 | 0 | 50 |
2 | 1 | 70 |
The GLIMMIX procedure is modeling the probability that mating='1'. |
There are two covariance parameters in this model, the variance of the fpop*fnum
effect and the variance of the mpop*mnum
effect (Output 43.2.4). Both parameters are modeled as G-side parameters. The nine columns in the matrix comprise the intercept, two columns each for the levels of the fpop
and mpop
effects, and four columns for their interaction. The matrix has 40 columns, one for each animal. Because the data are not processed by subjects, PROC GLIMMIX assumes the data
consist of a single subject (a single block in ).
Output 43.2.4: Model Dimensions Information
Dimensions | |
---|---|
G-side Cov. Parameters | 2 |
Columns in X | 9 |
Columns in Z | 40 |
Subjects (Blocks in V) | 1 |
Max Obs per Subject | 120 |
The “Optimization Information” table displays basic information about the optimization (Output 43.2.5). The default technique for GLMMs is the quasi-Newton method. There are two parameters in the optimization, which correspond to the two variance components. The 17 fixed effects parameters are not part of the optimization. The initial optimization computes pseudo-data based on the response values in the data set rather than from estimates of a generalized linear model fit.
Output 43.2.5: Optimization Information
Optimization Information | |
---|---|
Optimization Technique | Newton-Raphson with Ridging |
Parameters in Optimization | 2 |
Lower Boundaries | 2 |
Upper Boundaries | 0 |
Fixed Effects | Profiled |
Starting From | Data |
The GLIMMIX procedure performs eight optimizations after the initial optimization (Output 43.2.6). That is, following the initial pseudo-data creation, the pseudo-data were updated eight more times and a total of nine linear mixed models were estimated.
Output 43.2.6: Iteration History and Convergence Status
Iteration History | |||||
---|---|---|---|---|---|
Iteration | Restarts | Subiterations | Objective Function |
Change | Max Gradient |
0 | 0 | 4 | 537.09173501 | 2.00000000 | 1.719E-8 |
1 | 0 | 3 | 544.12516903 | 0.66319780 | 1.14E-8 |
2 | 0 | 2 | 545.89139118 | 0.13539318 | 1.609E-6 |
3 | 0 | 2 | 546.10489538 | 0.01742065 | 5.89E-10 |
4 | 0 | 1 | 546.13075146 | 0.00212475 | 9.654E-7 |
5 | 0 | 1 | 546.13374731 | 0.00025072 | 1.346E-8 |
6 | 0 | 1 | 546.13409761 | 0.00002931 | 1.84E-10 |
7 | 0 | 0 | 546.13413861 | 0.00000000 | 4.285E-6 |
Convergence criterion (PCONV=1.11022E-8) satisfied. |
The “Covariance Parameter Estimates” table lists the estimates for the two variance components and their estimated standard errors (Output 43.2.7). The heterogeneity (in the logit of the mating probabilities) among the females is considerably larger than the heterogeneity among the males.
Output 43.2.7: Estimated Covariance Parameters and Approximate Standard Errors
Covariance Parameter Estimates | ||
---|---|---|
Cov Parm | Estimate | Standard Error |
fpop*fnum | 1.4099 | 0.8871 |
mpop*mnum | 0.08963 | 0.4102 |
The “Type III Tests of Fixed Effects” table indicates a significant interaction between the male and female populations (Output 43.2.8). A comparison in the logits of mating success in pairs with R females and W females depends on whether the male partner in the pair is the same species. The “fpop*mpop Least Squares Means” table shows this effect more clearly (Output 43.2.9).
Output 43.2.8: Tests of Main Effects and Interaction
Type III Tests of Fixed Effects | ||||
---|---|---|---|---|
Effect | Num DF | Den DF | F Value | Pr > F |
fpop | 1 | 18 | 2.86 | 0.1081 |
mpop | 1 | 17 | 4.71 | 0.0444 |
fpop*mpop | 1 | 81 | 9.61 | 0.0027 |
Output 43.2.9: Interaction Least Squares Means
fpop*mpop Least Squares Means | ||||||||
---|---|---|---|---|---|---|---|---|
fpop | mpop | Estimate | Standard Error | DF | t Value | Pr > |t| | Mean | Standard Error Mean |
rb | rb | 1.1629 | 0.5961 | 81 | 1.95 | 0.0545 | 0.7619 | 0.1081 |
rb | ws | 0.7839 | 0.5729 | 81 | 1.37 | 0.1750 | 0.6865 | 0.1233 |
ws | rb | -1.4119 | 0.6143 | 81 | -2.30 | 0.0241 | 0.1959 | 0.09678 |
ws | ws | 1.0151 | 0.5871 | 81 | 1.73 | 0.0876 | 0.7340 | 0.1146 |
In a pairing with a male rough butt salamander, the logit drops sharply from 1.1629 to –1.4119 when the male is paired with a whiteside female instead of a female from its own population. The corresponding estimated probabilities of mating success are and . If the same comparisons are made in pairs with whiteside males, then you also notice a drop in the logit if the female comes from a different population, 1.0151 versus 0.7839. The change is considerably less, though, corresponding to mating probabilities of and . Whiteside females appear to be successful with their own population. Whiteside males appear to succeed equally well with female partners of the two populations.
This insight into the factor-level comparisons can be amplified by graphing the least squares mean comparisons and by subsetting the differences of least squares means. This is accomplished with the following statements:
ods graphics on; ods select DiffPlot SliceDiffs; proc glimmix data=salamander; class fpop fnum mpop mnum; model mating(event='1') = fpop|mpop / dist=binary; random fpop*fnum mpop*mnum; lsmeans fpop*mpop / plots=diffplot; lsmeans fpop*mpop / slicediff=(mpop fpop); run; ods graphics off;
The PLOTS=DIFFPLOT option in the first LSMEANS statement requests a comparison plot that displays the result of all pairwise comparisons (Output 43.2.10). The SLICEDIFF=(mpop fpop) option requests differences of simple effects.
The comparison plot in Output 43.2.10 is also known as a mean-mean scatter plot (Hsu, 1996). Each solid line in the plot corresponds to one of the possible unique pairwise comparisons. The line is centered at the intersection of two least squares means, and the length of the line segments corresponds to the width of a 95% confidence interval for the difference between the two least squares means. The length of the segment is adjusted for the rotation. If a line segment crosses the dashed 45-degree line, the comparison between the two factor levels is not significant; otherwise, it is significant. The horizontal and vertical axes of the plot are drawn in least squares means units, and the grid lines are placed at the values of the least squares means.
The six pairs of least squares means comparisons separate into two sets of three pairs. Comparisons in the first set are significant; comparisons in the second set are not significant. For the significant set, the female partner in one of the pairs is a whiteside salamander. For the nonsignificant comparisons, the male partner in one of the pairs is a whiteside salamander.
The “Simple Effect Comparisons” tables show the results of the SLICEDIFF= option in the second LSMEANS statement (Output 43.2.11).
Output 43.2.11: Simple Effect Comparisons
Simple Effect Comparisons of fpop*mpop Least Squares Means By mpop | |||||||
---|---|---|---|---|---|---|---|
Simple Effect Level |
fpop | _fpop | Estimate | Standard Error | DF | t Value | Pr > |t| |
mpop rb | rb | ws | 2.5748 | 0.8458 | 81 | 3.04 | 0.0031 |
mpop ws | rb | ws | -0.2312 | 0.8092 | 81 | -0.29 | 0.7758 |
Simple Effect Comparisons of fpop*mpop Least Squares Means By fpop | |||||||
---|---|---|---|---|---|---|---|
Simple Effect Level |
mpop | _mpop | Estimate | Standard Error | DF | t Value | Pr > |t| |
fpop rb | rb | ws | 0.3790 | 0.6268 | 81 | 0.60 | 0.5471 |
fpop ws | rb | ws | -2.4270 | 0.6793 | 81 | -3.57 | 0.0006 |
The first table of simple effect comparisons holds fixed the level of the mpop
factor and compares the levels of the fpop
factor. Because there is only one possible comparison for each male population, there are two entries in the table. The first
entry compares the logits of mating probabilities when the male partner is a rough butt, and the second entry applies when
the male partner is from the whiteside population. The second table of simple effects comparisons applies the same logic,
but holds fixed the level of the female partner in the pair. Note that these four comparisons are a subset of all six possible
comparisons, eliminating those where both factors are varied at the same time. The simple effect comparisons show that there
is no difference in mating probabilities if the male partner is a whiteside salamander, or if the female partner is a rough
butt. Rough butt females also appear to mate indiscriminately.