PSYC512: Research Methods Lecture 19

1. PSYC512: Research MethodsLecture 19 Brian P. Dyre University of Idaho PSYC512: Research Methods

2. Lecture 19 Outline • Inferential Statistics • Testing for differences vs. relationships • Analyzing frequencies • Analyzing differences between means PSYC512: Research Methods

3. Using Inferential Statistics • Which Statistic? • The statistical decision tree Howell Figure 1.1 • Testing for relationships vs. differences (a false distinction) • Relationships: assessing the strength of relationship between measured (dependent) variables • Differences: comparing different groups or treatments on some measurement • But what causes those differences? The relationship between the independent variable defining the groups or treatment and the dependent variable • Hence, testing for differences is really testing the relationship between the IV and DV PSYC512: Research Methods

4. Analyzing Differences Between Treatments • Nominal and Ordinal Frequency Data • “Success vs. Failure” - Binomial Distribution and The Sign Test • Multiple categories (> 2) Multinomial distribution and Chi-square • Multidimensional categories: Chi-square contingency tables • Integral and Ratio Data • 2 treatments or groups – t-test • Comparing two independent samples  HW3 • Comparing two correlated (or paired samples)  HW4 • More than 2 treatments or groups – ANOVA • More than 2 independent variables – multifactor ANOVA– HW5 • 2 or more dependent variables (or repeated measures) – MANOVA • Covariate  ANCOVA – HW5 • Relations between measures • Correlation or Regression PSYC512: Research Methods

5. Analyzing Frequencies (Howell, Chapter 5) • Bernoulli Trials: series of independent trials that result in one of two mutually exclusive outcomes • E.g. coin flips, gender of babies born, increase of decrease in a measure after application of a treatment • The Binomial Distribution PSYC512: Research Methods

6. Analyzing Frequencies (Howell, Chapter 5) • Using the binomial distribution • Mean number of successes = Np • Variance in number of successes = Npq • Testing Hypotheses using the binomial distribution: The Sign Test • Ho is typically p= q = .50 (50-50 chance of success of failure), but that doesn’t have to be the case • H1 is typically p ≠q • Plug in values for N, X, p, and q and p(X) directly provides the probability that the pattern of data could result given the null hypothesis is true • Sum the probabilities p(X) for all number >= X to get the total probability of finding p(>=X) • Important: The sign test takes into account direction of differences but not magnitude PSYC512: Research Methods

7. Analyzing Frequencies (Howell, Chapter 5) • What about multiple (more than 2) possible outcomes? • Multinomial distribution PSYC512: Research Methods

8. Analyzing Frequencies (Howell, Chapter 5) • Using the multinomial distribution • Mean Xk = NpXk • Variance in Xk = NpXk (1-pXk) • Testing Hypotheses using the multinomial distribution: • Ho is typically pX1= pX2 … = pXk = 1/k (each outcome has the same chance), but that doesn’t have to be the case • H1 is typically pX1 ≠ pX2 …≠ pXk • Plug in values for N, X, and pX, and p(X1, X2…Xk) directly provides the probability that this particular pattern of data could result given the null hypothesis is true • Must sum the probabilities for all patterns that deviate equal to or more to get the total probability – time consuming! PSYC512: Research Methods

9. Analyzing Frequencies (Howell, Chapter 6) • Easier Alternative to Multinomial distribution: Chi-square (c2) test • Compare computed value of c2 to value of c2 distribution with df=k-1 • Expected frequencies for the null hypothesis typically = N/k, where N is the total number of observations k is the number of categories in the variable O is the observed frequency for each category E is the expected frequency for each category i is the category index PSYC512: Research Methods

10. Analyzing Frequencies (Howell, Chapter 6) • Using c2 with multiple dimensions: contingency tables—frequencies of one dimension are contingent on the other dimension • Eij = RiCj/N • N is the total number of observations • Compare computed value of c2 to value of c2 distribution with df=(R-1)(C-1) R is the number of categories in the dimension defined by the rows of the table C is the number of categories in the dimension defined by the columns of the table O is the observed frequency for each category E is the expected frequency for each category i and j are category indices PSYC512: Research Methods

11. Analyzing Frequencies (Howell, Chapter 6) • Assumptions of the c2test • Each observation is independent • Inclusion of non-occurrences PSYC512: Research Methods

12. z-tests, t-tests • s of population is known: z • s of population is estimated as s: t • df = N-1 • Comparing 2 paired (or correlated) samples • Difference scores • Df = N -1 • Comparing 2 independent samples • df = n1 + n2 – 2 • Unequal sample sizes, heterogeneity of variance, and pooled variances PSYC512: Research Methods

13. ANOVA (F Statistic) • Used when comparing more than 2 means or 2 or more factors • Assumptions • Homogeneity of variance • Normality • Independence of observations • Between Groups comparisons • k = number of means compared • n = number of Ss in group • Repeated Measures • Error term is interaction of error with subject random variable PSYC512: Research Methods

14. Interpreting SPSS output PSYC512: Research Methods