Causality, Null Hypothesis Testing, and Bivariate Analysis P MGT 630 Statistics

Causality Theory - Some reason to logically believe or assume that x causes y. Time order - x comes before y in time. Correlation - When x change, y changes. Rule out alternative plausible explanations – Rule out or “control for” other reasons that x and y may correlate. For example, there may be one or more z variables that cause both x and y, making it appear that x and y correlate when they don’t

Null hypothesis testing Most statistical tests are based on the process of null hypothesis testing. A null hypothesis is the hypothesis that is being tested by the analysis. The p-value can be considered the probability that the null hypothesis is true. If the p-value is smaller than alpha (typically 0.05), then we say the null hypothesis is too improbable—we reject it in favor of the alternative.

Bivariate analysis Bivariate analysis is analysis that involves two variables. Some bivariate tests can be used to establish correlation. The outcome or effect variable (y) is called the dependent variable. The cause variable (x) is called the independent variable.

Selecting a test There are two main criteria for selecting an appropriate test: The null hypothesis The levels of measurement of the variables

Tests and Hypotheses

Paired value tests Null hypothesis: There is NO difference, on average, between the value in variable 1 and the value in variable 2. Frequent use: Pre-test and post-test comparisons (variable 1 is pre-test and variable 2 is post-test) Level of measurement assumptions: For interval variables, use paired t-test For ordinal variables, use Wilcoxon signed rank test

Two-group tests Null Hypothesis: The binary independent variable (group) does NOT affect the dependent variable. Frequent use: Determining whether the average value of the outcome variable differs based on (binary) group. Level of measurement assumptions: Independent variable is binary For interval dependent variable, use two group t-test For binary dependent variable, use two-group proportion test For ordinal dependent variable, use Wilcoxon-Mann-Whitney test

Chi-squared test Null Hypothesis: We would expect to observe the distribution of values by chance. Frequent use: Determining if there is something besides randomness determining the distribution of values in a table. Level of measurement assumptions: Dependent and independent variables are categorical Can also be used with binary variables, though it both variables are binary, there are more useful tests.

One-way ANOVA Null Hypothesis: The mean value for the dependent variable does NOT differ by group Common use: To determine whether or not one or more groups have different outcome values, on average. Level of measurement assumptions: Dependent variable is interval Independent variable is categorical

Kruskal-Wallis test Null Hypothesis: We would expect to observe the distribution of values by chance. Frequent uses: Determining if there is something besides randomness determining the distribution of values in a table that has an ordinal outcome variable. Level of measurement assumptions: The dependent variable is ordinal The independent variable is categorical

Regression Analysis Null Hypothesis: There is no correlation between x and y. Frequent use: To determine whether or not one variable correlates with another. Level of measurement assumptions Dependent variable is interval Categorical independent variables must be recoded as a set of binary (dummy) variables. If you include more than one dummy variables, this becomes multivariate analysis. If your independent variable is binary, the results will be identical to a two-group t-test.

Working with dummy variables When including a full set of dummy variables in a regression analysis, leave out one category to be the reference category. ORIGINAL Dummy1 Dummy2 Dummy3 Dummy4 1 2 3 4 5 (reference)

Logistic Regression Analysis Null Hypothesis: There is no correlation between x and y. Frequent use: To determine whether or not one variable correlates with another. Level of measurement assumptions Dependent variable is binary Categorical independent variables must be recoded as a set of binary (dummy) variables. If you include more than one dummy variables, this becomes multivariate analysis. If your independent variable is binary, compare results with the two-group proportion test.

Ordinal Logistic Regression Analysis Null Hypothesis: There is no correlation between x and y. Frequent use: To determine whether or not one variable correlates with another. Level of measurement assumptions Dependent variable is ordinal Categorical independent variables must be recoded as a set of binary (dummy) variables. If you include more than one dummy variables, this becomes multivariate analysis. If your independent variable is binary, compare results with the two-group proportion test.