3

Consider this example:

team <- rep(c("A","B","C"), times=c(7,4,10))
trip <- rep(NA,length(team))
for(i in 1:length(unique(team))){
  trip[which(team==unique(team)[i])] <- 1:days[i]
}
obs  < -c(rnorm(days[1],100,30), rnorm(days[1],100,5), rnorm(days[1],100,15))
data <- data.frame(team, trip, obs)

boxplot(obs~team, data)

It is pretty clear that the variance in each team is different, but the mean is similar.

How can I infer this statistically? How can I compare intra-group (within-group) variance with the inter-group (between-group) variance?

gung - Reinstate Monica
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IosuP
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  • I have no idea what you're asking/trying to do.. but intra group and inter group variance are related as follows. Var(Y)= E[Var(Y|X)]+Var(E[Y|X]) Interpretation: Let Y be player skill and X be dummies for teams. This says that the variance in player skill is the average within-team variance of skill plus the variance across teams of average within-team skill. IN your case, if the team means are similar, the second term is small. – CloseToC May 23 '14 at 13:48

2 Answers2

2

I'm not sure if I completely follow your thinking, but I can update this if it isn't what you're looking for.

In general we compare the intra-group variance with the inter-group variance with the ANOVA. (By the way, the more usual terms are within-group variance and between-group variance.) The standard use of the ANOVA may not be what you are after, though. It is used, ultimately, to determine if the means are the same (that's the inter-group variance part). In addition, it assumes that the group variances (intra-) are the same in order for the check of the inter-group variance to be valid.

If you want to know if the intra-group variances are the same, you can use Levene's test, which is an ANOVA on the absolute differences of each point from its group mean. In R, the function is leveneTest(obs, team, center=mean) in the car package (documentation). For a slightly more robust version, you can get the absolute value of the differences from the median and run an ANOVA on them instead. In that case, it is called the Brown-Forcythe test. That is actually the default for leveneTest (ironically), so you can just drop the center=mean part. I discuss these tests here: Why use Levene's test of equality of variance rather than F ratio?

If you believe you have substantial heteroscedasticity, but want to test if your means differ as well, there are a number of methods available. I discuss them here: Alternatives to one-way ANOVA for heteroscedastic data.

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gung - Reinstate Monica
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1

I'm not sure if this is what you're asking, but Levene's Test and a few similar tests in that family test if your groups have the same variance (test for homoscedasticity). So if Levene's test is significant, you reject the hypothesis that the variance of the populations you're sampling from is the same.

jona
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