Correlations and Regressions

Correlation

the degree of association between two variables: a pearsons correlation is continous

Regression

prediction of Y (the outcome variable/DV) on the basis of knowledge about X (predicitor variable/IV)

Scatterplot

useful to visualize relationships between 2 variables, the x axis is the predictor and the y axis is the outcome/criterion/DV

Line of best fit

shown in a graph of individual scores/points, degree to which points cluster around the line of best fit and ranges bewteen -1 and 1. This is a numerical quanitification for a correlation.

How many degrees of freedom do you subtract for a simple correlation?

Always 2 due to there being 2 variables we are making assumptions about

What are the factors that affect correlation?

Range restrictions, heterogenous subsamples, and covariance

Range restricitons

this is the most common thing that influences a correlation, results when variance in one or both variables is low, and results in a lowered correlation

Imagine a full scatterplot, and then drawing a square around a small piece of the scatterplot. You are missing the full picture!

Heterogenous subsamples

potential third variable problem, overall correlation may be artificially inflated or deflated because of two subsamples

Imagine a full scatterplot, and then drawing two separate boxes that splut the scatterplot. It is two DIFFERENT posible correlationbs within the graph.

Covariance: the statistical basis for correlation

reflects the degree to which two variables vary together, when one variable deviates from its mean we would expect the other variable to deviate from its own mean in a similar way

the main takeway to know: the equation for covariance examines how far scores are away from the mean, can be a positive or negative relationship.

Pearson product moment correlation coefficient (r)

This is used because the absolute value of the covariance is a function of the SDs of the variables, and standardizing covariance puts it in the understandable -1 to 1 window. USE PEOPLE FIRST LANGUAGE FOR APA.

SPSS for a simple correlation

Look for: This is a reflected table, the top and bottom are the same. The variables are on the top and bottoms, correlation first, p second, and N third in the box.

Wording: A Pearson correlation revealed that people who tend to report more control over thei life in general also tend to report more satisfaction with life, r(DF)= correlationcoe, p = siglvl.

Adjusted r

how meaningful, practicallt, less biased, used to get the less biased estimate of the population correlation coefficient, but in exchange you lose more DF, the difference between r adjusted and r will not be great for large samples but may be for small samples

The order of steps to make a correlation for one variable

Graph variables first, ensure there is variability, ask if there is normal dist, and ask if there are outliers and if they have undue influence on the results

Point biserial correlation (r_pb)

r where one is truly dichotomous and the other is approximately continous, is related to t, effect size is r_pb²

SPSS reading for Point Biserial

Look for: the table will say pearson because SPSS is evil, but it is point biserial. Same reflected type table as for pearson correlation.

Interpretation: A point biserial correlation revealed that there were no significant sex differences in self rated physical health, r_pb (DF) = pointbicorr, p = siglvl, r_pb² = percentvariance.

Spearmans corr coe

for ranked data (r_s), the significance terst can only be done if data cannot be normally distributed and there is no way to calculate standard error of (r_s) for a small n, a CI cannot be calculated

SPSS for Ranked data correlation

Look for: n > 10, reflected as with other correlation tables.

Interpretation: People who rank their own health as good (a low score on this scale) also tend to rank their health as better than their same age/sex peers (a high score), r_s (DF) = corrval, p = siglvl. NO EFFECT SIZE MEASURE FOR THIS.

Kendalls tau

for ranked data, based on the number of inversions/reversals in the rankings, should be used instead of spearmen when you have a small dataset with a large number of tied ranks, tau is a better rwtimate of the correlation in the pop than spearmen

SPSS for Kendall’s Tau

Look for: reflected table like other correlations.

Interpretation: Two versions.

1: Probability that the two variables will be ranked in this order is .43 hgiher than the probability that they will be ranked in the reverse order

2: People who rank their own health as good (a low score on this scale) also tend to rank their health as better than their same age/sex peers (a high score), Kendall’s Tau (DF) = corrval, p = siglvl. NO EFFECT SIZE MEASURE FOR THIS.

Regression

A regression is a line, and is Yhat = bX + a

Yhat = predicted value of Y

b = slope of regression line

a = intercept, when X = 0

X = predictor variable value

goal of a regression is to find the line of best fit that minimizes the error of squares, and has to pass though (0, a) and (X, Y)

What are standardized regression coeffecients

b becomes β if the data for each variable are standardized and if the differences in one unit of X or Y become one SD, r = slope if variables are standardized

Accuracy of prediciton

sum of squares is the sum of the SDs form a predicted point, standard error of the estimate is the sum of squares/df, residual/error variance = standard error of the estimate squared, and df is still n-2, there is an inverse relationship between r and standard error fo the estimate, r² is a measure of predicitve variability, like eta squared for correlations, does not relate to causation

Assumptions of regression and correlation

There is homogeneity of variance in arrays, variance of Y for each value of X is constant

There is a normality of arrays, the population values of Y corresponding to any particular value of X are normally distributed arounf Yhat

Confidence limits on Y

Can calculate the interval that 95% of the time will contain the value of Y for the given value of X, interval is smaller than what would be found if X was not used

Hypothesis testing

r for correlation

H_o : ρ = 0

t = (r (N-2)^(1/2))/(1-r²)^(1/2)

b (for each predictor in regression)

H_o = b* = 0, b* is when we assume the slope is flat

t = (b* - b) / standard error