stats week 8

stats week 8

1. Bivariate correlation

   • examines the relationship between 2 variables - relationships vary in form, (linear or curvilinear) - direction (positive/negative) - magnitude/strength ( r = - 1 perfect negative, r = + 1 perfect positive, r = 0 no relationship
     • weak correlation = 0.1 - 0.3, moderate = 0.4 - 0.6, strong = 0.7 - 0.9
   • non-linear correlation - relationship not best explained by straight line - best explained by a curve - would generate weak positive correlation because of straight line measuring , but is actually strong relationship

2. correlation

   • hypothesis testing - linear correlation involves measuring the realtionship between 2 variables measured in a sample, but we are actually interested in whether theres a relationship between the equivalent pop variables - use sample stats to estimate pop parameters
   • p-values - the probability of measuring a relationship of that magnitude when the null hypothesis is true

3. parametric linear correlation

   • Pearsons r (Pearsons product moment correlation) - assumptions and restrictions - data should be normal, very sensitive to outliers (sometimes appropriate to remove outliers) - sensitive to range restrictions - i.e floor and ceiling (all scores really low or high) effects - where data violate these assumptions or restrictions, you should use non-parametric correlations

4. covariance

   • how a score on one variable varies for a score on another variable - e.g mean IQ, and mean grade -
     • for each data point, calculate the difference from the mean of x, and the difference from the mean of y - multiple the differences, sum the multiplied differences , divide by N-1
   • provides a measure of the variance shared between our x and y variables
     • the correlation coefficient (r) is a ratio of the covariance to separate variances - we can obtain this measure of separate variances by multiplying the sanded deviation for x and y

5. correlation coefficient

   • if the covariance is large relative to the separate variances, r will be further from 0 - if the covariance is small relative to the separate variances, r will be closer to 0
     • r reflects how well a straight line fits the data points - (if the data points cluster closely to the line, r will be further from 0)

6. shared variance

   • r2 expresses the proportion of the separate variances that is shared (if r is 0.8, r2 is 0.64 - variables share 64%
     • r = 0.4, r2 = 0.16 - note that r = 0.8 is 4x as strong as r =0.4 (nots twice as strong)

7. effect size

   • r is a useful measure of effect size, it can be squared to give a measure of shared variance expressed as proportion of separate variances - it tells us how much variance in y can be explained by x

8. sampling distribution of correlations

   • all possible samples drawn from population of interest - those sampled correlation coefficients would build towards a normal distribution - null hypothesis states that there is no relationship between the population variables, so under the null the sampling distribution of correlation coefficients will have mean of 0
     • the r distribution has a mean of 0 - the extent to which an individual sampled correlation coefficient deviates from 0 can be expressed in SE units - using the r distribution, we can determine the prob of obtaining an r-value of a given magnitude when the null is true
       • confidence intervals - although we calculate r based on sample, we're interested in r of population - we can apply what we know about SND to give confidence intervals around our sample based estimate - SPSS doesn't provide confidence intervals around r, so we calculate by hand or use online calculator

9. partial correlation

   • allows us to examine relationship between 2 variables whilst removing influence of a third variable -

10. uses of correlations and non-parametric correlations

    • prediction - if there is a relationship between 2 variables we can make predictions about one from another - validity - concurrent validity (correlation between a new measure and established measure) - reliability - test-resort (are measures consistent) - inter rater reliability (are observers consistent) - theory verification - predictive validity
    • non parametric - scores are converted to ranks (fewer assumptions, robust to outliers, less sensitive) 0 spearmans rho (most common) - Kendalls tau (better option if you have fewer than 20 cases or many tied ranks