APPLIED DATA ANALYSIS IN CRIMINAL JUSTICE

CJ 525 MONMOUTH UNIVERSITY

Juan P. Rodriguez

Perspective Research Techniques Accessing, Examining and Saving Data Univariate Analysis – Descriptive Statistics Constructing (Manipulating) Variables Association – Bivariate Analysis Association – Multivariate Analysis Comparing Group Means – Bivariate Multivariate Analysis - Regression

Lecture 6

Comparing Group MeansBivariate Analysis

Relationships between categorical and numerical variables ANOVA:

Compares group means Test for significance

Bar Charts and Box Plots Tests for Differences in means

One Way ANOVA

How much the Mean Values of a Numerical Variable differ among the categories of a categorical variable

One Way ANOVA

Example: Relationship between television viewing and marital status in GSS98 dataset TVHOURS: numerical variable –

number of hours spent watching TV per day

MARITAL: categorical variable – married, widowed, divorced, separated and never married

One Way ANOVA

Null Hypothesis: No relationship - People in all

groups watch, on average, the same amount of television

Alternate Hypothesis: There is a relationship – At least 2

of the categories differ in the number of hours of television watched

Analysis Of Variance

Analysis Of Variance

Analysis Of Variance

Analysis Of Variance

Analysis Of Variance

Analysis Of Variance

Analysis Of Variance

Analysis Of Variance

Analysis Of Variance

Analysis Of Variance

Analysis Of Variance

Analysis Of Variance

The differences in the Mean values for these groups are so large that are not likely due to chance: There is a significant relationship

between marital status and television viewing

Graphing ANOVA Results

Bar charts Used to present data to general

people or to people not well versed in statistics

Box Plots Show both the central tendencies and

the distributions of each category

Bar Chart

Bar Chart

Bar Chart

Bar Chart

Bar Chart

Bar Chart

Bar Charts- Results

Separated and widowed people watch more TV, on the average, than the other categories of people

Box Plots

Depict differences in both the spread and center among groups of means.

By placing box plots side by side, it is easy to compare distributions

Box Plots

Box Plots

Box Plots

Box Plots

Box Plots

Box Plots

Post-hoc Tests ANOVA found significant differences

among means with respect to TV viewing

Are only 2 means significantly different?

Are all of them are significantly different?

Or anything in between?. Post-hoc tests tell us this

Post-hoc Tests

Post-hoc Tests

Post-hoc Tests

Post-hoc Tests

Post-hoc Tests

Post-hoc Tests

Post-hoc Tests

Post-hoc Tests

Post-hoc Tests

Assumptions in ANOVA Within each sample, the values are

independent, and identically normally distributed (same mean and variance).

The samples are independent of each other. The different samples are all assumed to come

from populations with the same variance, allowing for a pooled estimate of the variance.

For a multiple comparisons test of the sample means to be meaningful, the populations are viewed as fixed, so that the populations in the experiment include all those of interest.

Assumptions of ANOVA Distributions are normal:

The one-way ANOVA's F test is not affected much if the population distributions are skewed unless the sample sizes are seriously unbalanced.

If the sample sizes are balanced, the F test will not be seriously affected by light-tailedness or heavy-tailedness, unless the sample sizes are small (less than 5), or the departure from normality is extreme (kurtosis less than -1 or greater than 2).

In cases of nonnormality, a nonparametric test or employing a transformation may result in a more powerful test.

Assumptions of ANOVA Samples are Independent

A lack of independence within a sample is often caused by the existence of an implicit factor in the data.

Values collected over time may be serially correlated (here time is the implicit factor).

If the data are in a particular order, consider the possibility of dependence. (If the row order of the data reflect the order in which the data were collected, an index plot of the data [data value plotted against row number] can reveal patterns in the plot that could suggest possible time effects.)

Assumptions of ANOVA Variances are homogeneous:

Assessed by examination of the relative size of the sample variances, either informally (including graphically), or by a robust variance test such as Levene's test.

The risk of having unequal sample variances is incorrectly reporting a significant difference in the means when none exists. The risk is higher with greater differences between variances, particularly if there is one sample variance very much larger than the others.

Assumptions of ANOVA Variances are homogeneous (continued)

The F test is fairly robust against inequality of variances if the sample sizes are equal

If both nonnormality and unequal variances are present, use a transformation

A nonparametric test like the Kruskal-Wallis test still assumes that the population

variances are comparable.

Assumptions - Normality

Assumptions - Normality

Assumptions - Normality

Variance Homogeneity

Variance Homogeneity

Variance Homogeneity

Variance Homogeneity

t Tests

Compares the means of 2 groups Independent samples Paired Samples

t Test - Independent

t Test - Independent

t Test - Independent

t Test - Independent

t Test - Independent

t Test - Paired

Categories are related Are rates of incarceration the same

for black (PRC61) and whites (PRC58) in the states dataset?

Assumption is that states with high incarceration rates will tend to have high rates for blacks and whites

t Test - Paired

t Test - Paired

t Test - Paired

t Test - Paired

t Test - Paired

t Test - Paired