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The basic task of most research = Bivariate Analysis

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The basic task of most research = Bivariate Analysis

What does that involve?

Analyzing the interrelationship of 2 variables

Null hypothesis = independence (unrelatedness)

Two analytical perspectives:

Analysis of differences:

Select Independent Variable and Dependent variable

Compare Dependent Var. across values of Indep. Var.

Analysis of associations:

Covariation or Correspondence of variables

Predictability of one variable from the other

Agreement between two variables

“Bivariate Analysis”

Analytical situations:

If both variables = categorical?(either nominal or ordinal)

Use cross-tabulations (contingency tables) to show the relationship

If one variable (dependent)= categorical and other variable (independent) = numerical?

Use t-tests or ANOVA to test the relationship

What If both variables = numerical?

Then cross-tabs are no longer manageable and interpretable

T-tests and ANOVA don’t really apply

???

“Bivariate Analysis”

Analytical situations:

If both variables = numerical?

We can graph their relationship scatter plot

Need a statistical measure to index the inter-relationship between 2 numeric variables

This measure of the inter-relation of two numeric variables is called their “correlation”

“Bivariate Analysis”

Footnote: relevant questions about the relationship between variables

Does a relationship exist or are they independent? (significance test)

What is the form of the inter-relationship?

Linear or non-linear (for numerical variables)

Ordinal or Nonmonotonic (for ordinal variables)

Positive or negative (for ordered variables)

What is the strength of the relationship? (coefficient of association)

What is the meaning (or explanation) of the correlation? (not a statistical question)

I. Correlation

A quantitative measure of the degree of association between 2 numeric variables

The analytical model? Several alternative views:

Predictability

Covariance (mostly emphasizes this model)

I. Correlation

The analytical model for correlations:

Key concept = covariance of two variables

This reflects how strongly or consistently two variables vary together in a predictable way

Whether they are exactly or just somewhat predictable

It presumes that the relationship between them is “linear”

Covariance reflects how closely points of the bivariate distribution (of scores on X and corresponding scores on Y) are bunched around a straight line

Note the similarity with the formula for the variance of a single variable.

Correlation (continued)

Scatter Plot #1 (of moderate correlation):

Correlation (continued)

Scatter Plot #2 (of negative correlation):

Correlation (continued)

Scatter Plot #3 (of high correlation)

Correlation (continued)

Scatter Plot #4 (of very low correlation)

Correlation (continued)

How to compute a correlation coefficient?

By hand:

Definitional formula (the familiar one)

Computational formula (different but equivalent)

By SPSS: Analyze Correlate Bivariate

Correlation Coefficient (r): Definitional Formula

Correlation Coefficient (r): Computational Formula

Correlation (continued)

How to test correlation for significance?

Test Null Hypothesis that: r = 0

Use t-test:

Correlation (continued)

What are assumptions/requirements of correlation

Numeric variables (interval or ratio level)

Linear relationship between variables

Random sampling (for significance test)

Normal distribution of data (for significance test)

What to do if the assumptions do not hold

May be able to transform variables

May use ranks instead of scores

Pearson Correlation Coefficient (scores)

Spearman Correlation Coefficient (ranks)

Correlation (continued)

How to interpret correlations

Sign of coefficient?

Magnitude of coefficient ( -1 < r < +1)

Usual Scale: (slightly different from textbook)

+1.00 perfect correlation

+.75 strong correlation

+.50 moderately strong correlation

+.25 moderate correlation

+.10 weak correlation

.00 no correlation (unrelated)

-.10 weak negative correlation

(and so on for negative correlations)

Correlation (continued)

How to interpret correlations (continued)

NOTE: Zero correlation may indicate that relationShip is nonlinear (rather than no association between variables)

Important to check shape of distribution linearity; lopsidedness; weird “outliers”

Scatterplots = usual method

Line graphs (if scatter plot is hard to read)

May need to transform or edit the data:

Transforms to make variable more “linear”

Exclusion or recoding of “outliers”

Correlation (continued)

Scatterplots vs. Line graphs (example)

Correlation (continued)

crc319crc383dth177pvs500pfh493

crc319: Violent Crime rate -----.614-.048.268.034

crc383: Property Crime rate.614 -----.265.224.042

dth177: Suicide rate-.048.265 -----.178.304

pvs500: Poverty rate.268.224.178 ------.191

pfh493: Alcohol Consumption.034.042.304-.191 -----

How to report correlational results?

Single correlations (r and significance - in text)

Multiple correlations (matrix of coefficients in a separate table)

Note the triangular-mirrored nature of the matrix