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Nonparametric Statistics
nonparametric
statistics performed on nominal/ordinal dependent variables
do not require normality
Chi-square tests
nonparametric
^{}one of the most popular nonparametric tests
examines differences in proportions involving either one or two variables
One-way chi-square
nonparametric
involves only one varaible with 2 or more levels
Two-way chi-square
nonparametric
utilizes two variables with 2 or more levels
Chi-square assumptions
nonparametrc
1. independence of subjects
2. Not more than 20% of the expected frequencies is less than 5
Proportion
nonparametric
one of the simplest statistics that can be computed
number of responses (subjects) in a category divided by a total number of responses (subjects)
where n_{i }is the number of responses in the i^{th }category; N is the total number of responses
typically, the population is written as pi,
Properties of X^{2} Distribution
nonparametric
positively skewed distribution with ranges from 0 to positive infinity
-never negative
critical region of the distribution only appears in teh right side
-not a directional test
As df increases, the critical value of the distribution decreases
As df incerases, the distribution is less skewed
The mean of X^{2}^{ }distribution is equal to df
The variance of the X^{2}distribution is equal to the 2 x df
Chi-square Distribution
nonparametric
X^{2}tests have their own distribution
family of distributions denoted by a degrees of freedom
One-way X^{2
}^{nonparametric}
can be computed using either proportions or frequencies
-more common to use frequencies of categories
Example: Is there a difference in proportion of people who support and oppose abortion?
Two-wayX^{2
}^{nonparametric}
often seen as a test of independence (association)
both variables can be DVs or one variable can be a DV while the other is an IV
Cross Tabulation
nonparametric
Both variables are categorical variables with 2 or more levels
Variable 1 has c levels and variable 2 has r levels
r = row; c = column
O_{ij }will have a double subscript to indicate position in table
dot in 1st subscript = row total
dot in 2nd subscript = column total
Phi
nonparametric
used to measure effect size in a two way X^{2}
-can only be used in a 2 x 2
range from 0 to 1
similar interpretation as r
Cramer's V
nonparametric
used to measure effect size in a two way X^{2}
- larger than a 2 x 2 design
range from 0 to 1
similar interpretation as r
Assumptions
nonparametric
even though the assumption is on the expected frequencies, there is a higher chance of violating the assumption when the number of participants is small
Fisher's Exact Test
nonparametric
for a two-way X^{2}test with dichotomous variables (i.e. 2 X 2) where the assumption is most often violated
Yates' Correction for Continuity
nonparametric
when the test statistic becomes inaccurate
2 x 2
Cohen's w
nonparametric
w = phi
w = (Cramer's V)(square root of q)
Within-subject Design
nonparametric
(categorical)
Test of marginal homogeneity can be performed to examine a change in categorical variable from time 1 to 2 (within-subjects [repeated measures] design).