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...
![Page 1: Correlation and Regression A BRIEF overview Correlation Coefficients l Continuous IV & DV l or dichotomous variables (code as 0-1) n mean interpreted.](https://reader033.fdocuments.us/reader033/viewer/2022051619/56649dc55503460f94ab87f6/html5/thumbnails/1.jpg)
Correlation and Regression
A BRIEF overview
![Page 2: Correlation and Regression A BRIEF overview Correlation Coefficients l Continuous IV & DV l or dichotomous variables (code as 0-1) n mean interpreted.](https://reader033.fdocuments.us/reader033/viewer/2022051619/56649dc55503460f94ab87f6/html5/thumbnails/2.jpg)
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
![Page 3: Correlation and Regression A BRIEF overview Correlation Coefficients l Continuous IV & DV l or dichotomous variables (code as 0-1) n mean interpreted.](https://reader033.fdocuments.us/reader033/viewer/2022051619/56649dc55503460f94ab87f6/html5/thumbnails/3.jpg)
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
![Page 4: Correlation and Regression A BRIEF overview Correlation Coefficients l Continuous IV & DV l or dichotomous variables (code as 0-1) n mean interpreted.](https://reader033.fdocuments.us/reader033/viewer/2022051619/56649dc55503460f94ab87f6/html5/thumbnails/4.jpg)
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
![Page 5: Correlation and Regression A BRIEF overview Correlation Coefficients l Continuous IV & DV l or dichotomous variables (code as 0-1) n mean interpreted.](https://reader033.fdocuments.us/reader033/viewer/2022051619/56649dc55503460f94ab87f6/html5/thumbnails/5.jpg)
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
![Page 6: Correlation and Regression A BRIEF overview Correlation Coefficients l Continuous IV & DV l or dichotomous variables (code as 0-1) n mean interpreted.](https://reader033.fdocuments.us/reader033/viewer/2022051619/56649dc55503460f94ab87f6/html5/thumbnails/6.jpg)
Zero correlation
no relationship between the variables a positive or negative correlation gives
us predictive power
![Page 7: Correlation and Regression A BRIEF overview Correlation Coefficients l Continuous IV & DV l or dichotomous variables (code as 0-1) n mean interpreted.](https://reader033.fdocuments.us/reader033/viewer/2022051619/56649dc55503460f94ab87f6/html5/thumbnails/7.jpg)
Direction and degree
![Page 8: Correlation and Regression A BRIEF overview Correlation Coefficients l Continuous IV & DV l or dichotomous variables (code as 0-1) n mean interpreted.](https://reader033.fdocuments.us/reader033/viewer/2022051619/56649dc55503460f94ab87f6/html5/thumbnails/8.jpg)
Direction and degree (cont.)
![Page 9: Correlation and Regression A BRIEF overview Correlation Coefficients l Continuous IV & DV l or dichotomous variables (code as 0-1) n mean interpreted.](https://reader033.fdocuments.us/reader033/viewer/2022051619/56649dc55503460f94ab87f6/html5/thumbnails/9.jpg)
Direction and degree (cont.)
![Page 10: Correlation and Regression A BRIEF overview Correlation Coefficients l Continuous IV & DV l or dichotomous variables (code as 0-1) n mean interpreted.](https://reader033.fdocuments.us/reader033/viewer/2022051619/56649dc55503460f94ab87f6/html5/thumbnails/10.jpg)
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 )(
![Page 11: Correlation and Regression A BRIEF overview Correlation Coefficients l Continuous IV & DV l or dichotomous variables (code as 0-1) n mean interpreted.](https://reader033.fdocuments.us/reader033/viewer/2022051619/56649dc55503460f94ab87f6/html5/thumbnails/11.jpg)
Raw score formula
])(][)([ 2222 YYNXXN
YXXYNr
![Page 12: Correlation and Regression A BRIEF overview Correlation Coefficients l Continuous IV & DV l or dichotomous variables (code as 0-1) n mean interpreted.](https://reader033.fdocuments.us/reader033/viewer/2022051619/56649dc55503460f94ab87f6/html5/thumbnails/12.jpg)
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
![Page 13: Correlation and Regression A BRIEF overview Correlation Coefficients l Continuous IV & DV l or dichotomous variables (code as 0-1) n mean interpreted.](https://reader033.fdocuments.us/reader033/viewer/2022051619/56649dc55503460f94ab87f6/html5/thumbnails/13.jpg)
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
![Page 14: Correlation and Regression A BRIEF overview Correlation Coefficients l Continuous IV & DV l or dichotomous variables (code as 0-1) n mean interpreted.](https://reader033.fdocuments.us/reader033/viewer/2022051619/56649dc55503460f94ab87f6/html5/thumbnails/14.jpg)
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
![Page 15: Correlation and Regression A BRIEF overview Correlation Coefficients l Continuous IV & DV l or dichotomous variables (code as 0-1) n mean interpreted.](https://reader033.fdocuments.us/reader033/viewer/2022051619/56649dc55503460f94ab87f6/html5/thumbnails/15.jpg)
Correlation as causation??
![Page 16: Correlation and Regression A BRIEF overview Correlation Coefficients l Continuous IV & DV l or dichotomous variables (code as 0-1) n mean interpreted.](https://reader033.fdocuments.us/reader033/viewer/2022051619/56649dc55503460f94ab87f6/html5/thumbnails/16.jpg)
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
![Page 17: Correlation and Regression A BRIEF overview Correlation Coefficients l Continuous IV & DV l or dichotomous variables (code as 0-1) n mean interpreted.](https://reader033.fdocuments.us/reader033/viewer/2022051619/56649dc55503460f94ab87f6/html5/thumbnails/17.jpg)
Linear vs. curvilinear relationships
![Page 18: Correlation and Regression A BRIEF overview Correlation Coefficients l Continuous IV & DV l or dichotomous variables (code as 0-1) n mean interpreted.](https://reader033.fdocuments.us/reader033/viewer/2022051619/56649dc55503460f94ab87f6/html5/thumbnails/18.jpg)
Linear vs. curvilinear (cont.)
![Page 19: Correlation and Regression A BRIEF overview Correlation Coefficients l Continuous IV & DV l or dichotomous variables (code as 0-1) n mean interpreted.](https://reader033.fdocuments.us/reader033/viewer/2022051619/56649dc55503460f94ab87f6/html5/thumbnails/19.jpg)
Range restriction
![Page 20: Correlation and Regression A BRIEF overview Correlation Coefficients l Continuous IV & DV l or dichotomous variables (code as 0-1) n mean interpreted.](https://reader033.fdocuments.us/reader033/viewer/2022051619/56649dc55503460f94ab87f6/html5/thumbnails/20.jpg)
Range restriction (cont.)
![Page 21: Correlation and Regression A BRIEF overview Correlation Coefficients l Continuous IV & DV l or dichotomous variables (code as 0-1) n mean interpreted.](https://reader033.fdocuments.us/reader033/viewer/2022051619/56649dc55503460f94ab87f6/html5/thumbnails/21.jpg)
Understanding r
![Page 22: Correlation and Regression A BRIEF overview Correlation Coefficients l Continuous IV & DV l or dichotomous variables (code as 0-1) n mean interpreted.](https://reader033.fdocuments.us/reader033/viewer/2022051619/56649dc55503460f94ab87f6/html5/thumbnails/22.jpg)
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
![Page 23: Correlation and Regression A BRIEF overview Correlation Coefficients l Continuous IV & DV l or dichotomous variables (code as 0-1) n mean interpreted.](https://reader033.fdocuments.us/reader033/viewer/2022051619/56649dc55503460f94ab87f6/html5/thumbnails/23.jpg)
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
![Page 24: Correlation and Regression A BRIEF overview Correlation Coefficients l Continuous IV & DV l or dichotomous variables (code as 0-1) n mean interpreted.](https://reader033.fdocuments.us/reader033/viewer/2022051619/56649dc55503460f94ab87f6/html5/thumbnails/24.jpg)
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
![Page 25: Correlation and Regression A BRIEF overview Correlation Coefficients l Continuous IV & DV l or dichotomous variables (code as 0-1) n mean interpreted.](https://reader033.fdocuments.us/reader033/viewer/2022051619/56649dc55503460f94ab87f6/html5/thumbnails/25.jpg)
Moving to prediction
statistically significant relationship between college entrance exam scores and GPA
how can we use entrance scores to predict GPA?
![Page 26: Correlation and Regression A BRIEF overview Correlation Coefficients l Continuous IV & DV l or dichotomous variables (code as 0-1) n mean interpreted.](https://reader033.fdocuments.us/reader033/viewer/2022051619/56649dc55503460f94ab87f6/html5/thumbnails/26.jpg)
Best-fitting line (cont.)
![Page 27: Correlation and Regression A BRIEF overview Correlation Coefficients l Continuous IV & DV l or dichotomous variables (code as 0-1) n mean interpreted.](https://reader033.fdocuments.us/reader033/viewer/2022051619/56649dc55503460f94ab87f6/html5/thumbnails/27.jpg)
Best-fitting line (cont.)
![Page 28: Correlation and Regression A BRIEF overview Correlation Coefficients l Continuous IV & DV l or dichotomous variables (code as 0-1) n mean interpreted.](https://reader033.fdocuments.us/reader033/viewer/2022051619/56649dc55503460f94ab87f6/html5/thumbnails/28.jpg)
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
![Page 29: Correlation and Regression A BRIEF overview Correlation Coefficients l Continuous IV & DV l or dichotomous variables (code as 0-1) n mean interpreted.](https://reader033.fdocuments.us/reader033/viewer/2022051619/56649dc55503460f94ab87f6/html5/thumbnails/29.jpg)
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 )(
![Page 30: Correlation and Regression A BRIEF overview Correlation Coefficients l Continuous IV & DV l or dichotomous variables (code as 0-1) n mean interpreted.](https://reader033.fdocuments.us/reader033/viewer/2022051619/56649dc55503460f94ab87f6/html5/thumbnails/30.jpg)
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