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Correlation tests:

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Correlation Coefficient: A succinct measure of the strength of the relationship between two variables (e.g. height and weight, age and reaction time, IQ and exam score).

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There are various types of correlation coefficient, for different purposes: 1. Pearson's "r": Used when both X and Y variables are (a) continuous; (b) (ideally) measurements on interval or ratio scales; (c) normally distributed - e.g. height, weight, IQ. 2. Spearman's rho: In same circumstances as (1), except that data need only be on an ordinal scale - e.g. attitudes, personality scores.

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r is a parametric test: the data have to have certain characteristics (parameters) before it can be used. rho is a non-parametric test - less fussy about the nature of the data on which it is performed.

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Correlations vary between: +1 (perfect positive correlation: as X increases, so does Y):

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... and -1 (perfect negative correlation: as X increases, Y decreases, or vice versa). r = 0 means no correlation between X and Y: changes in X are not associated with systematic changes in Y, or vice versa.

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Calculating Pearson's r: a worked example: Is there a relationship between the number of parties a person gives each month, and the amount of flour they purchase from Vinny Millar?

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Month:Flour production (X): No. of parties (Y): X2X2 Y2Y2 XY A3775136956252775 B4178168160843198 C4888230477444224 D3280102464002560 E3678129660842808 F307190050412130 G4075160056253000 H4583202568893735 I3974152154762886 J3474115654762516 N=10ΣX = 382ΣY =776ΣX 2 = 14876ΣY 2 = 60444ΣXY = 29832

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10 776 60444 10 382 14876 10 776382 29832 r 22 Using our values (from the bottom row of the table:) N=10ΣX = 382ΣY =776ΣX 2 = 14876ΣY 2 = 60444ΣXY = 29832

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7455. 391.253 80.188 40.22660.283 80.188 r 60.602176044440.1459214876 20.2964329832 r r is.75. This is a positive correlation: people who buy a lot of flour from Vinny Millar also hold a lot of parties (and vice versa).

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How to interpret the size of a correlation: r 2 is the "coefficient of determination". It tells us what proportion of the variation in the Y scores is associated with changes in X. e.g., if r is.2, r 2 is 4% (.2 *.2 =.04 = 4%). Only 4% of the variation in Y scores is attributable to Y's relationship with X. Thus, knowing a person's Y score tells you essentially nothing about what their X score might be.

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Our correlation of.75 gives an r 2 of 56%. An r of.9, gives an r 2 of (.9 *.9 =.81) = 81%. Note that correlations become much stronger the closer they are to 1 (or -1). Correlations of.6 or -.6 (r 2 = 36%) are much better than correlations of.3 or -.3 (r 2 = 9%), not merely twice as strong!

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Spearman's rho: Measures the degree of monotonicity rather than linearity in the relationship between two variables - i.e., the extent to which there is some kind of change in X associated with changes in Y: Hence, copes better than Pearson's r when the relationship is monotonic but non-linear - e.g.:

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Spearman's rho - worked example: Is there a correlation between the number of vitamin treatments a person has, and their score on a memory test?

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Subj:No.vitamin teatments (X): Memory test score (Y): Vitamin treatment ranks (X): Memory ranks (Y): D (= X-Y)D2D2 A22221+11 B134121 C3363.53+0.50.25 D4495500 E3423.54-0.50.25 F65776+11 G58267.5-1.52.25 H88287.5+0.50.25 N = 8 ΣD 2 = 6.0

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OR

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Step 1: assign ranks to the raw data, for each variable separately. Rules for ranking: (a) Give the lowest score a rank of 1; next lowest a rank of 2; etc. (b) If two or more scores are identical, this is a "tie": give them the average of the ranks they would have obtained had they been different. The next score that is different, gets the rank it would have had if the tied scores had not occurred.

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e.g.: raw score1215151617 "original"rank12345 actual rank:12.52.545 Rank for the tied scores is (2+3)/2 = 2.5 raw score3181818100 "original"rank12345 actual rank:13335 Rank for the tied scores is (2+3+4)/3 = 3

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Step 2: Subtract one set of ranks from the other, to get a set of differences, D. Step 3: Square each of these differences, to get D 2. Step 4: Add up the values of D 2, to get ΣD 2. Here, ΣD 2 = 6.0 N = 8.

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Step 5:

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rho =.93. There is a strong positive correlation between the number of vitamin treatments a person has, and their memory test score. Pearson's r on the same data =.86.

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Using SPSS/PASW to obtain scatterplots: (a) simple scatterplot: Graphs > Legacy Dialogs > Scatter/Dot...

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Using SPSS/PASW to obtain scatterplots: (a) simple scatterplot: Graphs > Chartbuilder 1. Pick ScatterDot 2. Drag "Simple scatter" icon into chart preview window. 3. Drag X and Y variables into x-axis and y-axis boxes in chart preview window

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Using SPSS/PASW to obtain scatterplots: (b) scatterplot with regression line: Analyze > Regression > Curve Estimation... "Constant" is the intercept, "b1" is the slope

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Using SPSS/PASW to obtain correlations: Analyze > Correlate > Bivariate...

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