PSYC 60: Chapter 13 Study Guide
Correlation
A relationship between variables -A change in one variable is associated with a concurrent change in the other variable -Does NOT identify cause & effect relationships -"Correlation does not imply causation" -A can cause B -B can cause A -C can cause A & B
A positive correlation reveals that the variables tend to have similar values. A negative correlation reveals that the variables tend to have opposing values.
Absolute value of the correlation reveals the *strength* of the relationship between the two variables. The more extreme the r value (i.e., the farther it is from 0), the stronger the relationship. A coefficient of 1 or −1 is the strongest association that is possible.
Computational Formula Logic
Allow you to compute an r value directly from raw scores without first converting everything to z scores; this saves a lot of time.
Negative Data on Scatterplot?
Because the data points are also perfectly organized into a straight line, but the line trends from the *upper left to the lower right*.
Positive Data on Scatterplot?
Because the data points suggest a trend from the *lower left to the upper right side*.
r
Correlation Coefficient
Assumptions for Correlation and Regression
Data Independence -one pair scores doesn't affect the other Appropriate Variables -both interval=Pearson's or at least one must be ordinal=spearman's Normality -each variable has a normal shape in the population Scatterplot has -liner trend=Pearson's -monotonic trend=spearman's
Homoscedasticity
Data are homoscedastic if the amount of variability in the data is similar across the entire scatterplot.
Which of the following are correlations designed to accomplish?
Determine if two variables are associated Quantify the degree or strength of the association between two variables.
Symbolic & Verbal Rep. of One Tailed H1 and Ho for Pearson's Correlation
H1= p>0 (gratitude and prosocial attitudes are positively related) Ho= p≤0 (gratitude and prosocial attitudes are not positively related)
Symbolic & Verbal Rep. of Two-Tailed H1 and Ho for Pearson's Correlation
H1= p≠0 (gratitude and prosocial attitudes are linearly related; the correlation is not zero) Ho= p=0 (gratitude and prosocial attitudes are not linearly related; the correlation is zero)
Summary on Pearson & Spearman's correlation
If both variables are interval or ratio, you should use a Pearson's correlation by analyzing the raw data, but if either variable is ordinal, you need to use a Spearman's correlation by converting both variables to ranks and then analyzing those ranks.
When to not use Pearson's correlation?
If one or both of the variables are measured on an ordinal scale, you cannot use a Pearson's correlation. However, Spearman's correlation is very similar to Pearson's correlation, and it can analyze ordinal data.
Hypothesis Testing With Correlation
If the null were true, the calculated r value would be close to 0. If the calculated r value is far from 0, the null is not likely to be true.
A negative correlation between two variables indicates that high scores on one variable are associated with ______ scores on the second variable.
Low
Which of the following statements is true about statistics and causation?
No statistics, on their own, are sufficient evidence for inferring causality.
If both variables being analyzed are measured on an interval or ratio scale, a _______ correlation should be used.
Pearson's
REGRESSION
Regression is used to predict one score from another -Those variables must be correlated first -Strong correlation = More accurate prediction
What type of graph do we use for correlation?
Researchers frequently use *scatterplot* graphs to examine the relationship between variables.
If one of the variables is measured on an ordinal scale and the other variable is measured on an ordinal, interval, or ratio scale, a __________ correlation should be used.
Spearman's
If the trend revealed by a scatterplot is not linear but it is monotonic, what correlation should be used?
Spearman's
One-Tailed Pearson's Correlation Example
Step 1: Assess Statistical Assumptions Step 2: State the Null and Research Hypotheses Symbolically and Verbally Step 3: Define the Critical Region Step 4: Compute the Test Statistic (Pearson's r) Step 5: Compute the Effect Size (r2) and Describe It Step 6: Summarize the Results
Two-Tailed Pearson's Correlation
Step 1: Assumptions Step 2: State Null and Research Hypotheses Symbolically and Verbally Step 3: Define the Critical Region Step 4: Compute the Test Statistic (Pearson's r) Step 5: Compute the Effect Size (r2) and Describe It Step 6: Summarize the Results
Summarizing Results
The APA reporting format for correlations is similar to that used for other statistics; however, with other statistics, an effect size estimate is included in the summary statement. *ex:* There is a positive association between gratitude and prosocial attitudes, r(8) = .65, p < .05. That is, those with higher gratitude scores also tended to have more positive attitudes toward helping others.
How do we know if the relationship is strong?
The data points form an "organized" line rather than a more haphazard cluster. As the data points in a scatterplot "scatter" from a perfectly organized line, the strength of the relationship between the two variables gets weaker.
Correlation coefficients can vary between −1 and +1.
The sign of the coefficient reveals the direction of the variables' relationship.
A two-tailed correlation has ______ critical region(s).
Two (+ & -)
Computational Formula in Correlation
You need to compute; SSxy, ΣXY, ΣX, ΣY, and N. SSxy = ΣXY-(ΣX-ΣY)/N then plug SSxy into r= SSxy/√SSxSSy
In a scatterplot, each dot represents
a set of paired X and Y scores.
Positive z scores represent values that are
above average
The _______ of a z score indicates how much better or worse a given score is than the mean score.
absolute value
A correlation can only be used if the scores on each variable
are paired or linked to each other in some way.
When computing statistics by hand, researchers most often use _________ formulas. When trying to understand how a statistic works, students will find it easier to work with _________ formulas. However, both formulas will always produce ________ value(s).
computational; definitional; the same
In an APA-style reporting statement, the number in parentheses after r is
df
Critical Region One Tailed
df = N - 2
Critical Region Two Tailed
df = N − 2 N= # of paired scores
Use a two-tailed critical value when the research hypothesis
does not predict a specific direction for the relationship between the two variables.
Summarizing One Tailed Results
ex: The students' gratitude scores and prosocial attitudes were not significantly correlated, r(8) = .46, p > .05. However, the medium to large association between the variables suggests that the null hypothesis may not have been rejected because the sample size (N = 10) was too small. A larger sample size is needed to study the relationship between these variables.
A negative correlation coefficient means that
if someone has a high score on one variable, he or she tends to have a low score on the second variable.
If the research hypothesis predicts a positive association or a negative association, use a one-tailed test.
if the research hypothesis does not predict a specific direction, only that the two variables will be related somehow, use a two-tailed test.
Computing the Effect Size (r2) and Describe It
*formula:* r^2 The size of a correlation is described by r^2, which is also called the "coefficient of determination." The coefficient of determination is interpreted as a percentage. Specifically, r^2 = .42 indicates that 42% of the variability in prosocial attitudes is predicted by the variability in gratitude scores.
PLOTTING A REGRESSION LINE
-Regression line will always pass between (MeanX, MeanY) -Need 1 other point Choose any X and calculate Ŷ -Put those two points on your scatterplot and connect
General Guidelines for interpreting r^2
.01=Small .09=Medium .25=Large
If the null hypothesis is true and two variables are not associated with each other, the r value should be close to...
0
If two variables have a weak relationship, the absolute value of the correlation coefficient will be close to
0
The research hypothesis for a two-tailed Pearson's correlation predicts that the test statistic r will be far from
0
When, on average, paired z scores tend to have opposite signs, the resulting r value is
negative
When, on average, both paired z scores tend to be positive or both paired z scores tend to be negative, the resulting r value is...
positive
If r^2 = .36, it means that 36% of the variability in one variable is
predicted by the variability in the other variable.
If an obtained r value is farther from 0 than the critical value, you should
reject the null hypothesis.
Effect Size one Tailed
same as before, *formula:* r^2
Pearson's r (r) Correlations
use Pearson's r when both variables are measured on interval or ratio scales.
Spearman's (rs) Correlations
use Spearman's rs when one or both variables are measured in ordinal scale.
Computing the Test Statistic (Pearson's r)
using computational formula; SSxy = ΣXY-(ΣX-ΣY)/N then plug SSxy into r= SSxy/√SSxSSy
Formula's to Know for regression
Ŷ= bX + a b= r* SDy/SDx a= My - (b * Mx)
Regression Formula
Ŷ=bX + a •"Y hat" = predicted score -b= slope of the regression line -X = value used to predict Y -a = Y-intercept (predicted Y when X = 0)
