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.

**SAS**

In SAS, we'll read the data in with a simple input statement.

data milk;To fit the model, we'll use the

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;

`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 EstimatesThere'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

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 0.1623

`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 EffectsSo there's some reason to suspect that the suppliers may be different.

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 0.0563

**R**

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)To fit the model with unequal variances, we'll use the

mc = as.factor(rep(1:4, each=4))

milk= data.frame(value, mc)

`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)To assess the hypothesis of equal variances, we'll compare to the homoscedasticity model using the

mod = gls(value~mc, data=milk, weights = varIdent(form = ~1|mc),

method="ML")

`anova()`function.

mod2 = gls(value~mc, data=milk, method="ML")

anova(mod,mod2)

Model df AIC BIC logLik Test L.Ratio p-valueThis result is identical to SAS, although this is a likelihood ratio test and SAS shows a Wald test.

mod 1 8 125.3396 131.5203 -54.66981

mod2 2 5 124.4725 128.3355 -57.23625 1 vs 2 5.132877 0.1623

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")With ML in both programs we get the same results.

anova(mod3,mod)

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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