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Multivariate regression: MANOVA table

Simultaneous confidence intervals

Sometimes, we want to look at the differences between the means of a variable, for two different subsets. For example, we may want to test whether the mean age of male patients or female patients is different. One way to do this — to test the equality, according to two different types — is to look at the confidence interval for the difference between the respect means. In other words, we can look at a null hypothesis and hypothesis,

Or, alternatively,

To calculate the confidence interval, our multiplier will be the univariate t-statistic. However, we have to take into consideration the number of variables and the number of groupings,




t for probability and degrees of freedom

For example, for a 95% significance level, then our .


t-multiplier probability

t-multiplier degrees of freedom

Spreadsheet formula

= ABS (T.INV ( / (p * g * (g - 1)) , n - g))

So for example, if we have a sample with 4 variables, 3 groupings, and 516 observations, then, we have =T.INV(0.05/(4*3*2),513) which gives us a t-multiplier of 2.878.


Number of variables.


Number of groupings. We calculate g as the number of groupings (subsets) that exist. For example, if we are testing a generic version of a drug, then we may have samples for a placebo, the brand name version, and the generic version. In that case, the number of groupings is .


Number of observations across all groupings being considered. If we are looking at just two groupings, then our n is the number of observations for those two groupings, rather than the total number of observations for all groupings.

The number of observations in subgrouping . Similarly, there are observations in subgrouping , and so forth.

Once we have these components figured out, we can calculate the standard error for the confidence interval.

Standard error


To calculate W, we need the covariance matrix for each grouping . Then, we combine them,

In other words, the separate covariances are pooled together according to the respective numbers of observations.

When we are comparing the means for a variable for two groupings, then we will want the pooled variance for that variable, . Remember, we are not looking at two variables, we are looking at one variable, as observed in the groupings (subsets) within the sample.

mean difference of variable

The mean difference is calculated as,

Confidence interval

The confidence interval for the difference between the means between two groupings (subsets) within the sample, with one grouping called and one grouping called , is calculated as,