Our data today comes from a real example of vitamin D supplementation of milk. Four suppliers claimed that their milk provided 100 IU of vitamin D. The null hypothesis is that they all deliver this accurately, but there was some question about whether the variance was the same between the suppliers. Unfortunately, there are only four observations for each supplier.
In SAS, we'll read the data in with a simple input statement.
data milk; input value milk_cat; cards; 77 1 85 1 91 1 88 1 93 2 101 2 126 2 103 2 103 3 88 3 109 3 85 3 95 4 83 4 91 4 86 4 ;; run;To fit the model, we'll use the group option to the repeated statement in proc mixed. This is specifically designed to allow differing values for groups sharing the same covariance structure. In this case, it's a simple structure: no correlation, constant value on the diagonal. The key pieces of output are selected out using ODS. To assess variance terms, it's best to use maximum likelihood, rather than REML fitting.
ods select covparms lrt; proc mixed data = milk method = ml; class milk_cat; model value = milk_cat/solution; repeated/group=milk_cat type = simple; run;
Covariance Parameter Estimates Cov Parm Group Estimate Residual milk_cat 1 27.1875 Residual milk_cat 2 150.69 Residual milk_cat 3 100.69 Residual milk_cat 4 21.1875 Null Model Likelihood Ratio Test DF Chi-Square Pr > ChiSq 3 5.13 0.1623There's not too much evidence to support different variances, but the power is likely quite small, so we'll retain this model when assessing the null hypothesis of equal means. A REML fit ought to be preferable here, but to agree with the R results, we fit again with ML. Note that proc mixed is not smart enough to accurately determine the degrees of freedom remaining (16 observations - 4 variance parameters - 4 mean parameters) so we need to manually specify the denominator degrees of freedom using the ddf option to the model statement.
ods select solutionf tests3; proc mixed data = milk method = ml; class milk_cat; model value = milk_cat/solution ddf=8; repeated/group=milk_cat type = simple; run;
Solution for Fixed Effects Standard Effect milk_cat Estimate Error DF t Value Pr > |t| Intercept 88.7500 2.3015 12 38.56 <.0001 milk_cat 1 -3.5000 3.4776 8 -1.01 0.3437 milk_cat 2 17.0000 6.5551 8 2.59 0.0319 milk_cat 3 7.5000 5.5199 8 1.36 0.2113 milk_cat 4 0 . . . . Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F milk_cat 3 8 3.86 0.0563So there's some reason to suspect that the suppliers may be different.
In R, we'll re-type this simple data set and assign the group labels manually.
value = c(77,85,91,88,93,101,126,103,103,88,109,85,95,83,91,86) mc = as.factor(rep(1:4, each=4)) milk= data.frame(value, mc)To fit the model with unequal variances, we'll use the gls() function in the nlme library. (Credit department: Ben Bolker discusses this and more complex models in an Rpubs document.) (Note that we use maximum likelihood, as we did in SAS.)
library(nlme) mod = gls(value~mc, data=milk, weights = varIdent(form = ~1|mc), method="ML")To assess the hypothesis of equal variances, we'll compare to the homoscedasticity model using the anova() function.
mod2 = gls(value~mc, data=milk, method="ML") anova(mod,mod2)
Model df AIC BIC logLik Test L.Ratio p-value mod 1 8 125.3396 131.5203 -54.66981 mod2 2 5 124.4725 128.3355 -57.23625 1 vs 2 5.132877 0.1623This result is identical to SAS, although this is a likelihood ratio test and SAS shows a Wald test.
To assess whether the suppliers are different, we need to compare to the model with just an intercept. When using REML for both models, there was a warning message and a different answer than SAS (using REML in SAS as well). So we'll stick with maximum likelihood here.
mod3 = gls(value~1,data=milk, weights = varIdent(form = ~1|mc), method="ML") anova(mod3,mod)With ML in both programs we get the same results.
Model df AIC BIC logLik Test L.Ratio p-value mod3 1 5 126.8858 130.7488 -58.44292 mod 2 8 125.3396 131.5203 -54.66981 1 vs 2 7.546204 0.0564
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