CATEGORICAL RESPONSE: COMPARING TWO PROPORTIONS:
Bivariate methods is the general category of statistical methods used when we have two variables. The outcome variable on which comparisons are made is called the response variable. The binary variable that specifies the groups is the explanatory variable. In an independent sample, observations in one sample are independent from observations in another sample. If two samples have the same subjects, they are dependent. If each subject in one sample is matched with a subject in another sample there are matched pairs and the data is dependent as well.

The formula for the standard error for comparing two proportions is:

A 95% confidence interval for the difference between two population proportions has the following formula:

The proportion (p̂) is called a pooled estimate, since it pools the total number of successes and total number of observations from two samples. This uses the presumption p1=p2. The test statistic uses the following formula:

The standard error for the test statistic uses the following formula:

QUANTITATIVE RESPONSE: COMPARING TWO MEANS:
The standard error for comparing two means has the following formula:

A 95% confidence interval for the difference between two population means has the following formula:

The confidence interval for the difference between two population means uses the t-distribution and not the z-distribution. Interpreting a confidence interval for the difference of means uses the following criteria:

1. Check whether or not 0 falls in the interval
   If it does, it could be that mean 1 is mean 2.
2. Positive confidence interval suggests that mean 1 – mean 2 is positive
   If the confidence interval only contains positive numbers, this suggests that mean 1 – mean 2 is positive. This suggests that mean 1 is larger than mean 2.
3. Negative confidence interval suggests that mean 1 – mean 2 is negative
   If the confidence interval only contains negative numbers, this suggests that mean 1 -  mean 2 is negative. This suggests that mean 1 is smaller than mean 2.
4. Group order is arbitrary
   It is arbitrary whether one group is group one or the other.

The test statistic of a significance test comparing two population means uses the following formula:

It uses minus zero because the null hypothesis is that there is no difference between the groups and is thus zero.

OTHER WAYS OF COMPARING MEANS AND COMPARING PROPORTIONS
If it is reasonable to expect that the variability as well as the mean is the same, under the null hypothesis, then the assumption is made that the standard deviations are the same. This is the F-test. This is not a robust test. It performs poorly if the populations are not close to normal.

The pooled standard deviation combines information from the two samples to provide a single estimate of variability and uses the following formula:

This method uses the t-distribution and the formula for the degrees of freedom when comparing two groups has the following formula:

The standard error uses the following formula when using the pooled standard deviation estimate:

 or 

The ratio of proportions for two groups is p̂1/p̂2. This ratio is also called the relative risk. A sample relative risk above 1 indicates an effect.

ANALYZING DEPENDENT SAMPLES
A benefit of using dependent samples is that sources of potential bias are controlled. For dependent samples, the difference between the means of the two samples equals the mean of the difference scores for the matched pairs. When there are dependent samples, we calculate the difference scores and then use the one-sample methods. The paired-difference t test uses the difference scores for the pairs of observations. Confidence intervals and two-sides tests are robust when using the paired-difference t test.

The z-test comparing proportions is called McNemar’s test. The test statistic equals the difference of yes/no and no/yes divided by the square root of their sum.

ADJUSTING FOR THE EFFECTS OF OTHER VARIABLES
A multivariate analysis uses more than two variables. The third variable in multivariate analysis is called the control variable. This is a variable that is held constant in a multivariate analysis. To analyse whether an association can be explained by a third variable, we treat that third variable as a control variable. Simpson’s paradox states that the direction of the association can change when you control for a variable. With statistical control, the results tend to change considerably when the control variable has a strong association both with the response variable and the explanatory variable.