Other tests of significance

Independent variables: continuous Dependent variable: continuous Correlation: Relationship between variables Regression: Correlation squared – proportion of change in the dependent variable that is accounted for by the change in the independent variable. Hypothesis: Height  Weight Simple height/weight r :.719. R 2:.52, meaning that 52% of the change in weight is accounted for by the change in height. The probability that one could get r and r 2 statistics this high if the null hypothesis is true is <.01 (less than 1 in 100). r 2 =.52** Correlation and Regression

Independent variables: Categorical or continuous Dependent variable: Categorical, dichotomous (e.g., “Yes/No, “M/F”, etc.) Hypothesis is that arresting domestic abusers reduces the risk that their victims will be assaulted in the future. IV’s are down the left. DV repeat victimization (Yes/No) is embedded. This table reports the influence of the independent variables on Repeat Victimization = Y. b is the logistic regression coefficient. (Exp b are the odds that correspond with a one-unit change in the IV. One is even odds or 50/50, less than 1 is lower odds, and more than 1 is higher odds.) For IV arrest, the negative sign indicates that arresting decreases repeat victimization, but not significantly. The other IV’s have positive, significant relationships with the DV – that repeat victimization is more likely (a single asterisk means that the probability that the null hypothesis of no relationship is true is less than 5 in 100.) Logistic regression

Independent variables: categorical Dependent variable: continuous An extension of the difference between the means t-test to additional groups Example: does officer professionalism vary between cities? (scale 1-10) Calculate the “F” statistic, look up the table. An “F” statistic that is sufficiently large can overcome the null hypothesis that the differences between the means are due to chance. Analysis of Variance CityL.A.S.F.S.D. Mean853

Stratify independent variable(s) F statistic is a ratio of “between-group” to “within” group differences. To overcome the null hypothesis, the differences in scores between groups (between cities and, overall, between genders) should be much greater than the differences in scores within cities Between group variance (error + systematic effects of ind. variable) Within group variance (how scores disperse within each city) Two-way analysis of Variance CityL.A.S.F.S.D. Mean – M1075 Mean - F632 Between Withi n

Reading tables

Hypothesis - Academic performance  Delinquency IV’s run down the left; DV Delinquency embedded in the table Different ways to measure the IV in the hypothesis (each is actually a separate independent variable) Additional, “control” independent variables. Unless other instructions are given, when evaluating the direction of the relationship assume that each means “Y” (Male = Y) or is scaled conventionally, low to high A “model is simply a different combination of independent variables Numbers in parentheses are usually the standard error Watch for direction of relationship! Positive means IV and DV go up and down together, negative means as one rises the other falls. * = prob. null hypo. true <.05 ** = prob. null hypo true <.01 *** = prob. null hypo true <.001

Delinquency measures Truancy Arrest Conviction Sometimes different measures of the dependent variable run across the top Hypothesis - Academic performance  Delinquency Each measure is actually a different dependent variable This example (it’s made up, has nothing to do with the earlier slide) indicates that GPA has a significant negative relationship with three measures of the DV: truancy, arrests and convictions. As GPA increases each of the others decreases. The strongest relationship is with truancy, where the probability that the null is true is less than one in a thousand (<.001)

Instead of using asterisks, sometimes the actual probability that the null hypothesis is true is given For significant relationships, look for probabilities of.05 or less (<.05)

Know where the independent and dependent variables are! Sometimes, like in this example, categories of the dependent variable run in rows, and the independent variable categories run in columns.

Final exam question on tables The final exam will ask the student to interpret a table. The hypothesis will be provided. To do so the student will have to find the independent variables The student will have to interpret the statistic that tests the relationship between the independent and dependent variables If there is at least one asterisk the null hypothesis of no relationship can be rejected. The more asterisks the less the probability that the null is true: – < means “less than” – * means the probability that the null is true is <.05, ** means that it is <.01, *** means that it is <.001 It is critical that the student recognize whether the statistic denotes a positive or negative relationship – Positive means that the independent and dependent variables rise and fall together – Negative means that as one rises the other falls The student will also be asked to interpret, in words, what happens to the relationship between the IV and DV as new categories of the dependent variable are introduced IMPORTANT: Interpret this table based strictly on the techniques you learned in this course. If the table supports a hypothesis that, say, wealth causes crime, then wealth causes crime!

Final exam preview

Estimating a population parameter from a sample statistic – Calculate the 95 percent confidence interval – the range of scores within which the parameter will fall to a 95 percent certainty (only 5 chances in 100 that the parameter falls outside this range.) Always use z of 1.96 in the calculation – Final exam question will require that you compute a confidence interval and interpret it using words Testing a hypothesis where the independent variable is categorical and the dependent variable is continuous – Use the difference between means test, which yields the “t” statistic – Final exam question will require that you compute a “t”, decide whether you can reject the null hypothesis and interpret the results using words Testing a hypothesis where the independent and dependent variables are both categorical – Use Chi-Square (X 2 ) – Final exam question will require that you place data in an “Observed” table, create an “Expected” table, compute X 2 and decide whether the results support the working hypothesis Interpreting a table – You will be presented a table and a hypothesis – You will be asked whether the data, taken as a whole, supports the working hypothesis – There will also be a word question that asks you to note what happens as new measures of the dependent variable are introduced Final exam questions