Chapter 9: Non-parametric Tests

Parametric vs Non-parametric

Chi-Square– 1 way– 2 way

Parametric Tests

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

Non-parametric Tests

Do not require normality Or interval level of measurement

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

Compares observed frequencies within groups to their expected frequencies.

HO = “observed” frequencies are not different from the “expected” frequencies.

Research hypothesis: They are different.

Chi Square Statistic

fo = observed frequency

fe = expected frequency

Chi Square Statistic

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( )f f

f

o e

e

One-way Chi Square

Calculate the Chi Square statistic across all the categories.

Degrees of freedom = k - 1, where k is the number of categories.

Compare value to Table of Χ2.

One-way Chi Square Interpretation If our calculated value of chi square is less

than the table value, accept or retain Ho

If our calculated chi square is greater than the table value, reject Ho

…as with t-tests and ANOVA – all work on the same principle for acceptance and rejection of the null hypothesis

Two-Way Chi Square

Review cross-tabulations (= contingency tables) from Chapter 2.

Are the differences in responses of two groups statistically significantly different?

One-way = observed vs expected Two-way = one set of observed

frequencies vs another set.

Two-way Chi Square

Comparisons between frequencies (rather than scores as in t or F tests).

So, null hypothesis is that the two or more populations do not differ with respect to frequency of occurrence.

rather than working with the means as in t test, etc.

Two-way Chi Square Example

Null hypothesis: The relative frequency [or percentage] of liberals who are permissive is the same as the relative frequency of conservatives who are permissive.

Categories (independent variable) are liberals and conservatives. Dependent variable being measured is permissiveness.

Two-Way Chi Square Example

Child-rearing Political OrientationPractices

Liberals Conservatives TotalPermissive 13 7 20

Non-permissive 7 13 20Total 20 20 40

Two-Way Chi Square Example

Because we had 20 respondents in each column and each row, our expected values in this cross-tabulation would be 10 cases per cell.

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

Child-rearing Political Orientation (Expected)Practices

Liberals Conservatives TotalPermissive 10 10 20

Non-permissive 10 10 20Total 20 20 40

Two-Way Chi Square Example

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

What frequencies would we see if there were no difference between groups (if the null hypothesis were true)?

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

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.

We’ll put the expected values in parentheses.

Two-Way Chi-Square Example

Political OrientationLiberals Conservatives Total

Permissive 15 (12.5) 10 (12.5) 25Not Permissive 5 (7.5) 10 (7.5) 15

Total 20 20 40

Two-Way Chi-Square Example

So the chi square statistic, from this data is

(15-12.5)squared / 12.5 PLUS the same values for all the other cells

= .5 + .5 + .83 + .83 = 2.66

Two-Way Chi-Square Example

df = (r-1) (c-1) , where r = rows, c =columns so df = (2-1)(2-1) = 1

From Table C, α = .05, chi-sq = 3.84

Compare: Calculate 2.66 is less than table value, so we retain the null hypothesis.

Chapter 9: Non-parametric Tests

Review Parametric vs Non-parametric Be able to calculate: Chi-Square (obs-exp2 ) / exp

– 1 way– 2 way

• (row total) x (column total) / N = expected value for that cell

• calculate chi-square and compare to table.