Correlation and Regression A BRIEF overview Correlation Coefficients l Continuous IV & DV l or...
-
Upload
samuel-randolph-wilcox -
Category
Documents
-
view
219 -
download
0
Transcript of Correlation and Regression A BRIEF overview Correlation Coefficients l Continuous IV & DV l or...
Correlation and Regression
A BRIEF overview
Correlation Coefficients
Continuous IV & DV or dichotomous variables (code as 0-1)
mean interpreted as proportion Pearson product moment correlation
coefficient range -1.0 to +1.0
Interpreting Correlations
1.0, + or - indicates perfect relationship 0 correlations = no association between
the variables in between - varying degrees of
relatedness r2 as proportion of variance shared by two
variables which is X and Y doesn’t matter
Positive Correlation
regression line is the line of best fit With a 1.0 correlation, all points fall
exactly on the line 1.0 correlation does not mean values
identical the difference between them is
identical
Negative Correlation
If r=-1.0 all points fall directly on the regression line
slopes downward from left to right sign of the correlation tells us the
direction of relationship number tells us the size or magnitude
Zero correlation
no relationship between the variables a positive or negative correlation gives
us predictive power
Direction and degree
Direction and degree (cont.)
Direction and degree (cont.)
Correlation Coefficient r = Pearson Product-Moment Correlation Coefficient zx = z score for variable x
zy = z score for variable y N = number of paired X-Y values Definitional formula (below)
N
zzr yx )(
Raw score formula
])(][)([ 2222 YYNXXN
YXXYNr
Interpreting correlation coefficients comprehensive description of relationship direction and strength need adequate number of pairs more than 30 or so same for sample or population population parameter is Rho (ρ) scatterplots and r more tightly clustered around line=higher correlation
Examples of correlations
-1.0 negative limit -.80 relationship between juvenile street
crime and socioeconomic level .43 manual dexterity and assembly line
performance .60 height and weight 1.0 positive limit
Describing r’s Effect size index-Cohen’s guidelines:
Small – r = .10, Medium – r = .30, Large – r = .50
Very high = .80 or more Strong = .60 - .80 Moderate = .40 - .60 Low = .20 - .40 Very low = .20 or less
small correlations can be very important
Correlation as causation??
Nonlinearity and range restriction
if relationship doesn't follow a linear pattern Pearson r useless
r is based on a straight line function if variability of one or both variables is
restricted the maximum value of r decreases
Linear vs. curvilinear relationships
Linear vs. curvilinear (cont.)
Range restriction
Range restriction (cont.)
Understanding r
Simple linear regression
enables us to make a “best” prediction of the value of a variable given our knowledge of the relationship with another variable
generate a line that minimizes the squared distances of the points in the plot
no other line will produce smaller residuals or errors of estimation
least squares property
Regression line
The line will have the form Y'=A+BX Where: Y' = predicted value of Y A = Y intercept of the line B = slope of the line X = score of X we are using to predict Y
Ordering of variables
which variable is designated as X and which is Y makes a difference
different coefficients result if we flip them generally if you can designate one as
the dependent on some logical grounds that one is Y
Moving to prediction
statistically significant relationship between college entrance exam scores and GPA
how can we use entrance scores to predict GPA?
Best-fitting line (cont.)
Best-fitting line (cont.)
Calculating the slope (b)
N=number of pairs of scores, rest of the terms are the sums of the X, Y, X2, Y2, and XY columns we’re already familiar with
22 )()(
))(()(
XXN
YXXYNb
Calculating Y-intercept (a)
b = slope of the regression line the mean of the Y values the mean of the X values
Y
X
XbYa )(
Let’s make up a small example
SAT – GPA correlation How high is it generally? Start with a scatter plot Enter points that reflect the relationship
we think exists Translate into values Calculate r & regression coefficients