Linear Correlation

PSYC 6130, PROF. J. ELDER 2 Perfect Correlation 2 variables x and y are perfectly correlated if they are related by an affine transform y = ax + b The correlation is positive if a>0 and negative if a<0. By corollary, 2 variables are perfectly positively correlated if and only if each pair of corresponding values has the same z-score. If the 2 variables are perfectly negatively correlated, corresponding z-scores will be equal in magnitude but opposite in sign.

PSYC 6130, PROF. J. ELDER 3 Pearson’s r

PSYC 6130, PROF. J. ELDER 4 Scatterplots

PSYC 6130, PROF. J. ELDER 5 Pearson’s r only measures linear dependence Two variables can have low correlation and still be highly dependent.

PSYC 6130, PROF. J. ELDER 6 Higher-Order Models

PSYC 6130, PROF. J. ELDER 7 Pearson’s r depends on the range of the variables under study r 2 measures the proportion of variance in one variable accounted for by the other. If the range of variable X is restricted, it will account for less of the variance in Y.

PSYC 6130, PROF. J. ELDER 8 Pearson’s r is Sensitive to Outliers Outlier (Fake Student)

PSYC 6130, PROF. J. ELDER 9 Standard Definition of Correlation (Population)

PSYC 6130, PROF. J. ELDER 10 Standard Definition of Correlation (Sample)

PSYC 6130, PROF. J. ELDER 11 Alternative (Equivalent) Formula

PSYC 6130, PROF. J. ELDER 12 Computational Formula covariance For a population: For a sample: unbiased covariance

PSYC 6130, PROF. J. ELDER 13 Example: 6130A Assignment Marks

End of Lecture 7 Wed, Oct

Correlation and the Power of Matched Tests

PSYC 6130, PROF. J. ELDER 16 Correlation and the Power of Matched t-tests Now that we understand correlation, we can better understand the power of matched t-tests when scores in the two conditions are correlated.

PSYC 6130, PROF. J. ELDER 17 Recall formulae for standard error for independent and matched tests Independent t-testMatched t-test

PSYC 6130, PROF. J. ELDER 18 Knowing the expected std error, we can estimate the expected t-value Independent t-testMatched t-test

PSYC 6130, PROF. J. ELDER 19 The power of matched t-tests Large positive correlations between scores in the two conditions will mean a greater expected t-score for the matched design. But keep in mind that the critical value for the matched design will be somewhat larger as well, due to a smaller df. Which test is more powerful is decided by the exact tradeoff between these two effects.

Applying Correlation Analysis

PSYC 6130, PROF. J. ELDER 21 Adjusted Correlation Coefficient

PSYC 6130, PROF. J. ELDER 22 Testing Pearson’s r for Significance

PSYC 6130, PROF. J. ELDER 23 Underlying Assumptions (For Inference) Independent random sampling Bivariate normal distribution Probability

PSYC 6130, PROF. J. ELDER 24 Applications of Pearson’s r Measuring reliability and validity –Examples: e.g., test-retest reliability Split-half reliability Inter-rater reliability Criterion validity of self-report (correlate self-report against behavioural measure) Correlation between tests that are supposed to measure the same thing. Correlation between algorithmic model and human responses in behavioural studies. Measuring relationships between variables (correlational studies) –e.g., frequency of cannabis and alcohol use Measuring relationships between IVs and DVs (experimental studies, when IV on interval/ratio scale –e.g., exam performance as a function of alcohol consumption on previous night.

PSYC 6130, PROF. J. ELDER 25 Power Analysis for Pearson’s r

PSYC 6130, PROF. J. ELDER 26 Confidence Intervals for Pearson’s r Pearson’s r is bounded on [-1..1]. Consequently, sampling distribution for r is not normal. Sampling distribution for  >0 is negatively skewed. Sampling distribution for  <0 is positively skewed. Thus confidence intervals are generally not symmetric.

PSYC 6130, PROF. J. ELDER 27 Fisher Transform Fisher transform (Appendix r′): Method for symmetrizing r to facilitate calculation of confidence interval using standard normal table.

PSYC 6130, PROF. J. ELDER 28 Confidence Intervals on r

End of Lecture 8 Nov

PSYC 6130, PROF. J. ELDER 30 Testing Difference of Pearson Correlations from 2 Independent Samples Converting the skewed r distribution to an (approximately) normal distribution allows straightforward two-sample testing:

PSYC 6130, PROF. J. ELDER 31 Example N=43 N=44