Design and Data Analysis in Psychology I

Salvador Chacón MoscosoSusana Sanduvete Chaves

School of PsychologyDpt. Experimental Psychology

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Lesson 11

Relationship between two quantitative variables

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INTRODUCTION When assumptions are accepted

(parametric tests): Simple linear regression (it is going to be

studied next academic year in the subject Design and Data Analysis in Psychology II).

Pearson correlation.

When assumptions are not accepted (non-parametric tests): Spearman correlation.

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PEARSON CORRELATION: DEFINITION

• rXY

• Coefficient useful to measure covariation between variables: in which way changes in a variable are associated to the changes in other variable.

• Quantitative variables (interval or ratio scale).• Linear relationship EXCLUSIVELY.• Values: -1 ≤ rXY ≤ +1.• Interpretation:

+1: perfect positive correlation (direct association).-1: perfect negative correlation (inverse association).0: no correlation.

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Perfect positive correlation: rxy = +1 (difficult to find in psychology)

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Positive correlation: 0 < rxy < +1

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Perfect negative correlation: rxy = -1 (difficult to find in psychology)

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Negative correlation: -1 < rxy < 0

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No correlation

Formulas

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YXXY SS

YXN

XY

r

22 yx

xyrXY

N

ZZr YXXY

Raw scores

Deviation scores

Standard scores

ExampleX: 2 4 6 8 10 12 14 16 18 20 Y:1 6 8 10 12 10 12 13 10 22

1. Calculate rxy in raw scores.

2. Calculate rxy in deviation scores.

3. Calculate rxy in standard scores.

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Example: scatter plot

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Example :calculation of rxy in raw scores

X Y XY X2 Y2

2 1 2 4 14 6 24 16 366 8 48 36 648 10 80 64 10010 12 120 100 14412 10 120 144 10014 12 168 196 14416 13 208 256 16918 10 180 324 10020 22 440 400 484110 104 1390 1540 1342

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Example :calculation of rxy in raw scores

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839.0103.5*745.5

4.10*1110

1390

YXXY SS

YXN

XY

r

1110

110

N

XX

4.1010

104

N

YY

745.51110

1540 222

XN

XSx

103.54.1010

1342 222

YN

YS y

Example :calculation of rxy in deviation scores

X Y x y xy x2 y2

2 1 -9 -9.4 84.6 81 88.364 6 -7 -4.4 30.8 49 19.366 8 -5 -2.4 12 25 5.768 10 -3 -0.4 1.2 9 0.1610 12 -1 1.6 -1.6 1 2.5612 10 1 -0.4 -0.4 1 0.1614 12 3 1.6 4.8 9 2.5616 13 5 2.6 13 25 6.7618 10 7 -0.4 -2.8 49 0.1620 22 9 11.6 104.4 81 134.56110 104 0 0 246 330 260.4

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Example :calculation of rxy in deviation scores

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839.04.260*330

24622

yx

xyrXY

Example :calculation of rxy in standard scores

X Y Zx Zy ZxZy2 1 -1.567 -1.842 2.8864 6 -1.218 -0.862 1.0516 8 -0.870 -0.470 0.4098 10 -0.522 -0.078 0.04110 12 -0.174 0.314 -0.05512 10 0.174 -0.078 -0.01414 12 0.522 0.314 0.16416 13 0.870 0.510 0.44318 10 1.218 -0.078 -0.09620 22 1.567 2.273 3.561110 104 0 0 8.391

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Example :calculation of rxy in standard scores

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839.010

391.8

N

ZZr YXXY

XX S

XXZ

YY S

YYZ

Significance

Does the correlation coefficient show a real relationship between X and Y, or is that relationship due to hazard?

Null hypothesis H0: rxy = 0. The correlation coefficient is drawn from a population whose correlation is zero (ρXY = 0).

Alternative hypothesis H1: . The correlation coefficient is not drawn from a population whose correlation is different to zero (ρXY ).

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0XYr

0

Significance Formula:

Interpretation: Null hypothesis is rejected. The

correlation is not drawn from a population whose score ρxy = 0. Significant relationship between variables exists.

Null hypothesis is accepted. The correlation is drawn from a population whose score ρxy = 0. Significant relationship between variables does not exist.

Exercise: conclude about the significance of the example.20

21 2

Nr

rt

XY

XY

)2,( Ntt

)2,( Ntt

Significance: example

Conclusions: we reject the null hypothesis with a maximum risk to fail of 0.05. The correlation is not drawn from a population whose score ρxy = 0. Relationship between variables exists. 21

37.4

210839.01

839.0

21 22

Nr

rt

XY

XY

306.2)8,05.0()2,( tt N

306.237.4

Other questions to be considered Correlation does not imply causality. Statistical significance depends on sample size (higher N,

likelier to obtain significance). Other possible interpretation is given by the coefficient of

determination , or proportion of variability in Y that is ‘explained’ by X.

The proportion of Y variability that left unexplained by X is called coefficient of non-determination:

Exercise: calculate the coefficient of determination and the coefficient of non-determination and interpret the results.

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2XYr

21 XYr

Coefficient of determination: example

70.4% of variability in Y is explained by X.

29.6% of variability in Y is not explained.

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704.0839.0 22 XYr

296.0839.011 22 XYr

Which is the final conclusion?

Significant effect

Non-significant effect

High effect size (≥ 0.67)

The effect probably exists

The non-significance can be

due to low statistical power

Low effect size (≤ 0.18)

The statistical significance can be due to an excessive

high statistical power

The effect probably does not exist

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Which is the final conclusion?

Significant effect

Non-significant effect

High effect size (≥ 0.67)

The effect probably exists

The non-significance can be

due to low statistical power

Low effect size (≤ 0.18)

The statistical significance can be due to an excessive

high statistical power

The effect probably does not exist

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