# Which is right? Individual variable t test or multivariate Hotelling t test?

So, lets say a hypothetical example.

You have data of women aged 25-50 on their nutrition activity and measured for 737 such women(n,sample size) on 5 variables(Calcium, Protein, Vitamin A, Vitamin B, Magnesium). Now lets say you have the recommended values by USDA and also have the mean values for each of these variables for your sample.

Now if all the variables are independent , then if you do an individual hypothesis testing(t-test) on each of the five variables then that procedure would be ok. But if there is correlation then doing a multivariate hypothesis test is right as it would take into account for correlation.

But now lets say, instead of having the whole 5 variables observations, there are two researchers. first one measures the calcium values on the same sample women and the other one measures Protein on the same sample women. They do individual hypothesis test and it turn out to be true(Null hypothesis: Sample values are same as recommended values and women are healthy). But then they meet and merge their datasets the joint hypothesis turns out negative as now it takes into account correlation between calcium and Protein intake and hence it affected the values and you reject the null hypothesis. Then which one is to be believed?.

My understanding says, joint hypothesis is right one to believe. Reason:

Hypothesis testing should be an inference about the whole population which should be acceptable by everyone. So individually to me it appears both researchers are not using the full information to come at a conclusion hence their inference is biased(crudely sample biased as they only have information on one measure). But a joint one is more applicable. What is the opinion of others here.

• I'm not sure I understand the reason to add the first paragraph and mention the 5 variables? Why not just use your last 3 paragraphs as a question? Am I missing something here about the other 3 measures? Also are you trying to ask something about the measurements being taken by two different researchers? I'm not sure your questions is very clear to the community. If you can clean it up and make it a bit more succinct and to the point, I'm guessing you'll get some pretty reasonable answers. – StatsStudent Mar 12 '15 at 22:01
• By the way the multivariate test is testing something different than the univariate tests. Using two separate tests amounts to performing independent tests of $H_1$: $\mu_{calcium}=\mu_{c0}$ and then separately, $H_2$: $\mu_{protein}=\mu_{p0}$. The Hotelling $T^2$ test tests whether or not the entire vector $X$=$[\mu_{calcium}, \mu_{protein}]$ = $[\mu_{c0}, \mu_{p0}]$. One would typically proceed by performing the multivariate test and then if significant, moving on to individual tests. – StatsStudent Mar 12 '15 at 22:06
• Your situation, as described, has nothing to do with t-tests or Hotelling's T squared; it would be appropriate for univariate vs multiple regression. You may find the answer you want here. You should also read the threads categorized under multicollinearity to learn more about the effects of correlated predictors. – gung - Reinstate Monica Mar 16 '15 at 3:23