Chapter 9: Non-parametric Tests n Parametric vs Non-parametric n Chi-Square –1 way –2 way

Parametric Tests n Data approximately normally distributed. n Dependent variables at interval level. n Sampling random n t - tests n ANOVA

Non-parametric Tests n Do not require normality n Or interval level of measurement n Less Powerful -- probability of rejecting the null hypothesis correctly is lower. So use Parametric Tests if the data meets those requirements.

One-Way Chi Square Test n Compares observed frequencies within groups to their expected frequencies. n H O = “observed” frequencies are not different from the “expected” frequencies. n Research hypothesis: They are different.

Chi Square Statistic n f o = observed frequency n f e = expected frequency

Chi Square Statistic

One-way Chi Square n Calculate the Chi Square statistic across all the categories. n Degrees of freedom = k - 1, where k is the number of categories. n Compare value to Table of Χ 2.

One-way Chi Square Interpretation n If our calculated value of chi square is less than the table value, accept or retain H o n If our calculated chi square is greater than the table value, reject H o n …as with t-tests and ANOVA – all work on the same principle for acceptance and rejection of the null hypothesis

Two-Way Chi Square n Review cross-tabulations (= contingency tables) from Chapter 2. n Are the differences in responses of two groups statistically significantly different? n One-way = observed vs expected n Two-way = one set of observed frequencies vs another set.

Two-way Chi Square n Comparisons between frequencies (rather than scores as in t or F tests). n So, null hypothesis is that the two or more populations do not differ with respect to frequency of occurrence. n rather than working with the means as in t test, etc.

Two-way Chi Square Example n Null hypothesis: The relative frequency [or percentage] of liberals who are permissive is the same as the relative frequency of conservatives who are permissive. n Categories (independent variable) are liberals and conservatives. Dependent variable being measured is permissiveness.

Two-Way Chi Square Example

n Because we had 20 respondents in each column and each row, our expected values in this cross-tabulation would be 10 cases per cell. n Note that both rows and columns are nominal data -- which could not be handled by t test or ANOVA. Here the numbers are frequencies, not an interval variable.

Two-Way Chi Square Expected

Two-Way Chi Square Example n Unfortunately, most examples do not have equal row and column totals, so it is harder to figure out the expected frequencies.

Two-Way Chi Square Example n What frequencies would we see if there were no difference between groups (if the null hypothesis were true)? n If 25 out of 40 respondents(62.5%) were permissive, and there were no difference between liberals and conservatives, 62.5% of each would be permissive.

Two-Way Chi Square Example n We get the expected frequencies for each cell by multiplying the row marginal total by the column marginal total and dividing the result by N. n We’ll put the expected values in parentheses.

Two-Way Chi-Square Example

n So the chi square statistic, from this data is n ( )squared / 12.5 PLUS the same values for all the other cells n = = 2.66

Two-Way Chi-Square Example n df = (r-1) (c-1), where r = rows, c =columns so df = (2-1)(2-1) = 1 n From Table C, α =.05, chi-sq = 3.84 n Compare: Calculate 2.66 is less than table value, so we retain the null hypothesis.

Chapter 9: Non-parametric Tests n Review Parametric vs Non-parametric n Be able to calculate: n Chi-Square (obs-exp 2 ) / exp –1 way –2 way (row total) x (column total) / N = expected value for that cell calculate chi-square and compare to table.