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

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

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

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

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

PSYC512: Research Methods Analyzing Frequencies (Howell, Chapter 5) Using the binomial distribution Using the binomial distribution Mean number of successes = Np Mean number of successes = Np Variance in number of successes = Npq Variance in number of successes = Npq Testing Hypotheses using the binomial distribution: The Sign Test 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 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 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 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) 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 Important: The sign test takes into account direction of differences but not magnitude

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

PSYC512: Research Methods Analyzing Frequencies (Howell, Chapter 5) Using the multinomial distribution Using the multinomial distribution Mean X k = Np Xk Mean X k = Np Xk Variance in X k = Np Xk (1-p Xk ) Variance in X k = Np Xk (1-p Xk ) Testing Hypotheses using the multinomial distribution: Testing Hypotheses using the multinomial distribution: Ho is typically p X1 = p X2 … = p Xk = 1/k (each outcome has the same chance), but that doesn’t have to be the case Ho is typically p X1 = p X2 … = p Xk = 1/k (each outcome has the same chance), but that doesn’t have to be the case H1 is typically p X1 ≠ p X2 … ≠ p Xk H1 is typically p X1 ≠ p X2 … ≠ p Xk Plug in values for N, X, and p X, and p(X 1, X 2 …X k ) directly provides the probability that this particular pattern of data could result given the null hypothesis is true Plug in values for N, X, and p X, and p(X 1, X 2 …X k ) 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! Must sum the probabilities for all patterns that deviate equal to or more to get the total probability – time consuming!

PSYC512: Research Methods Analyzing Frequencies (Howell, Chapter 6) Easier Alternative to Multinomial distribution: Chi-square (    test Easier Alternative to Multinomial distribution: Chi-square (    test Compare computed value of   to value of   distribution with df=k-1 Compare computed value of   to value of   distribution with df=k-1 Expected frequencies for the null hypothesis typically = N/k, where N is the total number of observations 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 Analyzing Frequencies (Howell, Chapter 6) Using   with multiple dimensions: contingency tables—frequencies of one dimension are contingent on the other dimension Using   with multiple dimensions: contingency tables—frequencies of one dimension are contingent on the other dimension E ij = R i C j /N E ij = R i C j /N N is the total number of observations N is the total number of observations Compare computed value of   to value of   distribution with df=(R-1)(C-1) Compare computed value of   to value of   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 Assumptions of the    test Assumptions of the    test Each observation is independent Each observation is independent Inclusion of non-occurrences Inclusion of non-occurrences Analyzing Frequencies (Howell, Chapter 6)

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

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

PSYC512: Research Methods Interpreting SPSS output