Linear Function Y a bX Fixed and Random Variables A FIXED variable is one for which you have every possible value of interest in your sample Example Subject sex female or male A RANDOM variable is one where the sample values are randomly obtained from the population of values ID: 533516
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Slide1
Bivariate Linear CorrelationSlide2
Linear Function
Y = a + bXSlide3
Fixed and Random Variables
A FIXED variable is one for which you have every possible value of interest in your sample.
Example: Subject sex, female or male.
A RANDOM variable is one where the sample values are randomly obtained from the population of values.
Example: Height of subject.Slide4
Correlation & Regression
If Y is random and X is fixed, the model is a regression model.
If both Y and X are random, the model is a correlation model.
Psychologists generally do not know this
They think
Correlation = compute the corr coeff,
r
Regression = find an equation to predict Y from XSlide5
Scatter PlotSlide6Slide7Slide8Slide9Slide10
For the data plotted below, the linear
r
= 0, but the quadratic
r
= 1.Slide11
Burgers (X) and Beer (Y)Slide12
Burger (X)-Beer (Y) Correlation
.Slide13Slide14Slide15Slide16
Burger (X)-Beer (Y) Correlation
.Slide17
H
ø
:
ρ = 0
df
=
n
– 2 = 3
Now
get an exact
p
value and construct
a confidence intervalSlide18
Get Exact p
Value
COMPUTE p=2*CDF.T(t,df).Slide19
Go To Vassar
http://vassarstats.net/
Slide20Slide21
N increased to 10.Slide22
Presenting the Results
The correlation between my friends’ burger consumption and their beer consumption fell short of statistical significance,
r
(
n
= 5) = .8,
p
= .10,
95% CI [-.28, .99].
Among my friends, beer consumption was positively, significantly related to burger consumption,
r
(
n
= 10) = .8,
p
= .006,
95% CI [.
34,
.
95].Slide23
Assumptions
Homoscedasticity across Y|X
Normality of Y|X
Normality of Y ignoring X
Homoscedasticity across X|Y
Normality of X|Y
Normality of X ignoring Y
The first three should look familiar, we made them with the pooled variances
t
.Slide24
Bivariate NormalSlide25
When Do Assumptions Apply?
Only when employing t or
F
.
That is, obtaining a
p
value
or constructing a confidence interval.Slide26
Shrunken r
2
This reduces the bias in estimation of
As sample size increases (
n
-1)/(
n
-2) approaches 1, and the amount of correction is reduced.Slide27
Do not use Pearson
r if the relationship is not linear. If it is monotonic, use Spearman rho.Slide28
Every time X increases, Y decreases – accordingly we have here
a perfect, negative, monotonic relationshipSlide29Slide30
Pearson r
measures the strength of the linear relationship. Notice that it is NOT perfect here.Slide31
Spearman rho measures the strength of monotonic relationship. Notice that it IS perfect here.Slide32
Uses of Correlation Analysis
Measure the degree of linear associationCorrelation
does
imply causation
Necessary but not sufficient
Third variable problems
Reliability
Validity
Independent Samples
t
– point biserial
r
Y = a + b
Group (Group is 0 or 1)Slide33
Uses of Correlation Analysis
Contingency tables --
Rows =
a
+
b
Columns
Multiple correlation/regressionSlide34
Uses of Correlation Analysis
Analysis of variance (ANOVA)
PolitConserv =
a
+
b
1
Republican? +
b
2
Democrat?
k
= 3, the third group is all others
Canonical correlation/regressionSlide35
Uses of Correlation Analysis
Canonical correlation/regression
(homophobia, homo-aggression) =
(psychopathic deviance, masculinity, hypomania, clinical defensiveness)
High homonegativity = hypomanic, unusually frank, stereotypically masculine, psychopathically deviant (antisocial)Slide36
Factors Affecting Size of r
Range restrictions
Without variance there can’t be covariance
Extraneous variance
The more things affecting Y (other then X), the smaller the
r
.
Interactions – the relationship between X and Y is modified by Z
If not included in the model, reduces the
r
.Slide37
Power AnalysisSlide38
Cohen’s Guidelines
.10 – small but not trivial.30 – medium.50 – largeSlide39
PSYC 6430 Addendum
The remaining slides cover material I do not typically cover in the undergraduate course.Slide40
Correcting for Measurement Error
If reliability is not 1, the
r
will underestimate the correlation between the latent variables.
We can estimate the correlation between the true scores this way:
r
xx
and
r
YY
are reliabilitiesSlide41
Example
r between misanthropy and support for animal rights = .36 among persons with an idealistic ethical ideology
Slide42
Comparing Correlation/Regression Coefficients
Weaver, B., & Wuensch, K. L. (2013).
SPSS and SAS programs for comparing Pearson correlations and OLS regression coefficients
.
Behavior Research Methods, 45
, 880-895.
doi
10.3758/s13428-012-0289-7Slide43
H
:
1
=
2
Is the correlation between X and Y the same in one population as in another?
The correlation between misanthropy and support for animal rights was significantly greater in nonidealists (
r
= .36) than in idealists (
r
= .02)Slide44
H
:
WX
=
WY
We have data on three variables. Does the correlation between X and W differ from that between Y and W.
W is GPA, X is SAT
verbal
, Y is SAT
math
.
See Williams’ procedure in our text.
See other procedures referenced in my handout.Slide45
H
:
WX
=
YZ
Raghunathan
, T. E, Rosenthal, R, & and Rubin, D. B. (1996). Comparing correlated but
nonoverlapping
correlations,
Psychological Methods
,
1
, 178-183.
Example: is the correlation between
verbal
aptitude
and math aptitude the same at 10 years of age as at twenty years of age (
longitudinal
data)Slide46
H
:
= nonzero value
A meta-analysis shows that the correlation between X and Y averages .39.
You suspect it is not .39 in the population in which you are interested.
H
:
= .39.