relationships between variables statistics for the social sciences psychology 340 spring 2010

Relationships between variables

Statistics for the Social SciencesPsychology 340

Spring 2010

PSY 340Statistics for the

Social Sciences Exam 3 results

• Mean = 86.9• Median = 89.0• Great job!


Social Sciences Final project

• Details posted on website• Download the Harass.sav datafile (fictional dataset

of harassment within a workplace)• Conduct analyses to answer 8 different questions• Write up the results of you analyses

• Worth 15% of final grade


Social SciencesOutline (for 2 weeks)

• Correlation– Scatterplot, hypothesis testing, computations, SPSS

• Simple bi-variate regression, least-squares fit line– The general linear model

– Residual plots

– Using SPSS

• Multiple regression– Comparing models, Delta r2

– Using SPSS


Social Sciences Correlation

• Correlations describe relationships between two variables

– Age and coordination skills in children, as kids get older their motor coordination tends to improve

– Price and quality, generally the more expensive something is the higher in quality it is



• Correlations describe relationships between two variables, but DO NOT explain why the variables are related

Suppose that Dr. Steward finds that rates of spilled coffee and severity of plane turbulents are strongly positively correlated.

One might argue that turbulents cause coffee spills

One might argue that spilling coffee causes turbulents




Suppose that Dr. Cranium finds a positive correlation between head size and digit span (roughly the number of digits you can remember).

One might argue that bigger your head, the larger your digit span

1

21

24

1537

One might argue that head size and digit span both increase with age (but head size and digit span aren’t directly related)




For many years instructors have noted that the reported fatality rate of

grandparents increases during midterm and final exam periods. One might argue that college exams cause grandparent death

Dead Grandmother/Exam Syndrome Web Page


Social Sciences Relationships between variables

• How variables co-vary with one another– As a descriptive statistic

• To examine this relationship you should:– Make a scatterplot - a picture of the relationship– Compute the Correlation Coefficient - a numerical description of the

relationship

• Properties of a correlation– Form (linear or non-linear)– Direction (positive or negative)– Strength (none, weak, strong, perfect)

– As an inferential statistic – comparing an observed correlation with a correlation expected due to chance


Social Sciences Scatterplot: Graphing Correlations

• Steps for making a scatterplot1. Draw axes and assign variables to

them

2. Determine range of values for each variable and mark on axes

3. Mark a dot for each person’s pair of scores

Hours

studied

Quiz

performance

A 6 6B 1 2C 5 6

D 3 4

E 3 2

X Y

Example: What is the relationship between how much you study and exam performance?


Social Sciences Scatterplot

Y

X1

2

34

5

6

1 2 3 4 5 6

• Plots one variable against the other• Each point

corresponds to a different individual

A 6 6

X Y

B 1 2C 5 6

D 3 4

E 3 2



Y

X1

2

34

5

6

1 2 3 4 5 6



A 6 6B 1 2

X Y

C 5 6

D 3 4

E 3 2



Y

X1

2

34

5

6

1 2 3 4 5 6



A 6 6B 1 2C 5 6

X Y

D 3 4

E 3 2



Y

X1

2

34

5

6

1 2 3 4 5 6



A 6 6B 1 2C 5 6

D 3 4

X Y

E 3 2



Y

X1

2

34

5

6

1 2 3 4 5 6



A 6 6B 1 2C 5 6

D 3 4

E 3 2

X Y



Y

X1

2

34

5

6

1 2 3 4 5 6

• Imagine a line through the data points



A 6 6B 1 2C 5 6

D 3 4

E 3 2

X Y

• Useful for “seeing” the relationship– Form, Direction,

and Strength


Social Sciences Form

Non-linearLinear


Social Sciences

NegativePositive

Direction

• X & Y vary in the same direction

• As X goes up, Y goes up

• Positive Pearson’s r

• X & Y vary in opposite directions

• As X goes up, Y goes down

• Negative Pearson’s r

Y

X

Y

X


Social Sciences Strength

• The strength of the relationship– Spread around the line (note the axis scales)

– Correlation coefficient will range from -1 to +1• Zero means “no relationship”

• The farther the r is from zero, the stronger the relationship


Social Sciences Strength

r = 1.0“perfect positive corr.”r2 = 100%

r = -1.0“perfect negative corr.”r2 = 100%

r = 0.0“no relationship”r2 = 0.0

-1.0 0.0 +1.0

The farther from zero, the stronger the relationship


Social Sciences Hypothesis testing with Pearson’s r

• Hypothesis testing– Core logic of hypothesis testing

• Considers the probability that the result of a study could have come about if the experimental procedure had no effect

• If this probability is low, scenario of no effect is rejected and the theory behind the experimental procedure is supported

• Step 1: State your hypotheses

• Step 2: Set your decision criteria

• Step 3: Collect your data

• Step 4: Compute your test statistics

• Step 5: Make a decision about your null hypothesis

– A five step program


Social Sciences

– Step 1: State your hypotheses: as a research hypothesis and a null hypothesis about the populations

• Null hypothesis (H0)

• Research hypothesis (HA)

Hypothesis testing with Pearson’s r

• There are no correlation between the variables (they are independent)

• Generally, the variables correlated (they are not independent)



r ≥

r <

H0:

HA:

– Our theory is that the variables are negatively correlated

– Step 1: State your hypotheses

One -tailed

Note: sometimes the

symbol ρ (rho) is used

Note: sometimes the

symbol ρ (rho) is used



r > 0

r < 0

H0:

HA:

– Our theory is that the variables are negatively correlated

– Step 1: State your hypotheses

One -tailed

r = 0

r ≠ 0

H0:

HA:

– Our theory is that the variables are correlated

Two -tailed



– Step 2: Set your decision criteria• Your alpha (α) level will be your guide for when to reject or fail

to reject the null hypothesis. – Based on the probability of making making an certain type of

error



– Step 3: Collect your data• Descriptive statistics (Pearson’s r)

6 61 25 6

3 4

3 2

X Y

• Common formulas for the correlation coefficient:

€

r =SP

SSX SSY

€

SP = X − X ( ) Y −Y ( )∑

Used this one in PSY138

r = XZ YZ∑N

Z-score alternative

For an example of the z-score alternative, skip to the end of the powerpoint


Social Sciences Computing Pearson’s r (using SP)

• Stage 1: SP (Sum of the Products)

€

SP = X − X ( ) Y −Y ( )∑

mean 3.6 4.0

6 61 25 6

3 4

3 2

X Y

€

X − X ( )

€

Y −Y ( )

€

X − X ( ) Y −Y ( )




€

SP = X − X ( ) Y −Y ( )∑

mean 3.6 4.0

2.4

0.0

6 61 25 6

3 4

3 2

X Y

€

X − X ( )

€

Y −Y ( )

€

X − X ( ) Y −Y ( )= 6 - 3.6

-2.6 = 1 - 3.6

1.4 = 5 - 3.6

-0.6 = 3 - 3.6

-0.6 = 3 - 3.6

Quick check




€

SP = X − X ( ) Y −Y ( )∑

mean 3.6 4.0

2.4-2.6

1.4

-0.6

-0.6

0.0 0.0

6 61 25 6

3 4

3 2

X Y

€

X − X ( )

€

Y −Y ( )

€

X − X ( ) Y −Y ( )2.0 = 6 - 4.0

-2.0 = 2 - 4.0

2.0 = 6 - 4.0

0.0= 4 - 4.0

-2.0= 2 - 4.0

Quick check




€

SP = X − X ( ) Y −Y ( )∑

mean 3.6 4.0

2.4-2.6

1.4

-0.6

-0.6

0.0

2.0-2.0

2.0

0.0

-2.0

0.0 14.0 SP

6 61 25 6

3 4

3 2

X Y

€

X − X ( )

€

Y −Y ( )

€

X − X ( ) Y −Y ( )4.8* =

5.2* =

2.8* =

0.0* =

1.2* =



• Stage 2: SSX & SSY




mean 3.6 4.0

2.4-2.6

1.4

-0.6

-0.6

0.0

2.0-2.0

2.0

0.0

-2.0

0.0 14.0

6 61 25 6

3 4

3 2

X Y

€

X − X ( )

€

Y −Y ( )

€

X − X ( ) Y −Y ( )4.85.2

2.8

0.0

1.2

€

X − X ( )2

5.76

15.20

SSX

2 =6.762 =

1.962 =

0.362 =

0.362 =




mean 3.6 4.0

2.4-2.6

1.4

-0.6

-0.6

0.0

2.0-2.0

2.0

0.0

-2.0

0.0 14.0

6 61 25 6

3 4

3 2

X Y

€

X − X ( )

€

Y −Y ( )

€

X − X ( ) Y −Y ( )4.85.2

2.8

0.0

1.2

€

X − X ( )2

5.766.76

1.96

0.36

0.36

15.20

€

Y −Y ( )2

2 = 4.02 = 4.02 = 4.02 = 0.02 = 4.0

16.0

SSY



• Stage 3: compute r

€

r =SP

SSX SSY




mean 3.6 4.0

2.4-2.6

1.4

-0.6

-0.6

0.0

2.0-2.0

2.0

0.0

-2.0

0.0 14.0

6 61 25 6

3 4

3 2

X Y

€

X − X ( )

€

Y −Y ( )

€

X − X ( ) Y −Y ( )4.85.2

2.8

0.0

1.2

€

X − X ( )2

5.766.76

1.96

0.36

0.36

15.20

€

Y −Y ( )2

4.04.0

4.0

0.0

4.0

16.0

SSYSSX

SP

€

r =SP

SSX SSY


Social Sciences Computing Pearson’s r


14.015.20 16.0

SSYSSX

SP

€

r =SP

SSX SSY




15.20 16.0

SSYSSX

r =14

SSXSSY




15.20

SSX

r =14

SSX * 16




€

r =14

15.2 *16




r =14

15.2 * 16=.89

Y

X1

2

34

5

6

1 2 3 4 5 6

• Appears linear

• Positive relationship

• Fairly strong relationship• .89 is far from 0, near +1



– Step 4: Compute your test statisticsr = 0.89 • Descriptive statistics (Pearson’s r)

• Inferential statistics: 2 choices (really the same):– A t-test & the t-table

– Use the Pearson’s r table (if available)

• Compute your degrees of freedom (df) df = n - 2 = 5 - 2 = 3



– Step 4: Compute your test statistics• Descriptive statistics (Pearson’s r)



t =r( ) n−2( )

1−r2=

.89( ) 3( )

1 − .79= 3.38

• From table, with df = n - 2 = 3: tcrit = 3.18

• Reject H0

• Conclude that the correlation is ≠0

– Step 5: Make a decision about your null hypothesis

r = 0.89


Social Sciences

Proportion in one tail 0.05 0.025 0.01 0.005

Proportion in two tails df 0.10 0.05 0.02 0.01 1 .988 .997 .9995 .9999 2 .900 .950 .980 .990 3 .805 .878 .934 .959 4 .729 .811 .882 .917 5 .669 .754 .833 .874 6 .622 .707 .789 .834 :

15 :

: .412

:

: .482

:

: .558

:

: .606

:

Hypothesis testing with Pearson’s r

– Step 4: Compute your test statistics

• From table– α-level = 0.05

– Two-tailed

– df = n - 2 = 3

– rcrit = 0.878

• Reject H0

• Conclude that the correlation is ≠0

– Step 5: Make a decision about your null hypothesis

• Descriptive statistics (Pearson’s r)



r = 0.89


Social Sciences Effect sizes with Pearson’s r

• Pearson’s r is considered a measure of the effect size – Small r = 0.10

– Medium r = 0.30

– Large r = 0.50


Social SciencesA few more things to consider about correlation

• Correlations are greatly affected by the range of scores in the data– Consider height and age relationship

• Extreme scores can have dramatic effects on correlations – A single extreme score can radically change r

• When considering "how good" a relationship is, we really should consider r2 (coefficient of determination), not just r.


Social Sciences Correlation in SPSS

• Enter each variable in separate columns– Analyze -> Correlate -> bi-variate

– Enter all variables you want to examine• In options can request cross products and means

– Output – given as a matrix

– For the scatterplot: • Graphs -> legacy dialogs-> scatter/dot -> simple scatter

• Enter which is your X var. and which is your Y var.


Social Sciences Correlation in Research Articles

• Correlation matrix– A display of the correlations between more than two

variablesAcculturation

• Why have a “-”?

• Why only half the table filled with numbers?


Social Sciences Next time

• Regression: Predicting a variable based on other variables


Social Sciences The Correlation Coefficient

• Formulas for the correlation coefficient:

r = XZ YZ∑N

r =SP

SSXSSY

SP = X−X( ) Y −Y( )∑

Used this one in PSY138 Common alternative


Social Sciences

Computing Pearson’s r (using z-scores)

• Step 1: compute standard deviation for X and Y (note: keep track of sample or population)

6 61 25 6

3 4

3 2

X Y

• For this example we will assume the data is from a population


Social Sciences



Mean 3.6

2.4-2.6

1.4

-0.6

-0.6

0.0

6 61 25 6

3 4

3 2

X Y

€

X − X ( )

€

X − X ( )2

5.766.76

1.96

0.36

0.36

15.20

SSXStd dev 1.74

σ =SSX

N=

15.2

5= 1.74



Social Sciences



Mean 3.6 4.0

2.4-2.6

1.4

-0.6

-0.6

2.0-2.0

2.0

0.0

-2.0

0.0

6 61 25 6

3 4

3 2

X Y X −X( )

€

Y −Y ( )X −X( )2

5.766.76

1.96

0.36

0.36

15.20

€

Y −Y ( )2

4.04.0

4.0

0.0

4.0

16.0

SSYStd dev 1.74 1.79


σ =SSY

N

=16.0

5= 1.79


Social Sciences


• Step 2: compute z-scores

Mean 3.6 4.0

2.4-2.6

1.4

-0.6

-0.6

2.0-2.0

2.0

0.0

-2.0

6 61 25 6

3 4

3 2

X Y

€

X − X ( ) Y −Y( )X −X( )2

5.766.76

1.96

0.36

0.36

15.20

Y −Y( )2

4.04.0

4.0

0.0

4.0

16.0Std dev

ZX

1.74 1.79

1.38 =2.4

1.74

X −X( )sX


Social Sciences



Mean 3.6 4.0

2.4-2.6

1.4

-0.6

-0.6

2.0-2.0

2.0

0.0

-2.0

6 61 25 6

3 4

3 2

X Y

€

X − X ( ) Y −Y( )X −X( )2

5.766.76

1.96

0.36

0.36

15.20

Y −Y( )2

4.04.0

4.0

0.0

4.0

16.0Std dev

ZX

X −X( )sX

1.74 1.79

1.38-1.49

0.8

- 0.34

- 0.34

0.0 Quick check


Social Sciences



Mean 3.6 4.0

2.4-2.6

1.4

-0.6

-0.6

2.0-2.0

2.0

0.0

-2.0

6 61 25 6

3 4

3 2

X Y X −X( )

€

Y −Y ( )X −X( )2

5.766.76

1.96

0.36

0.36

15.20

€

Y −Y ( )2

4.04.0

4.0

0.0

4.0

16.0Std dev

ZX ZY

1.74 1.79

1.1

Y −Y( )sY

=2.0

1.791.38-1.49

0.8

- 0.34

- 0.34


Social Sciences



Mean 3.6 4.0

2.4-2.6

1.4

-0.6

-0.6

2.0-2.0

2.0

0.0

-2.0

6 61 25 6

3 4

3 2

X Y X −X( )

€

Y −Y ( )X −X( )2

5.766.76

1.96

0.36

0.36

15.20

€

Y −Y ( )2

4.04.0

4.0

0.0

4.0

16.0Std dev

ZX ZY

Y −Y( )sY

1.74 1.79

1.1-1.1

0.0

-1.1

1.1

0.0

1.38-1.49

0.8

- 0.34

- 0.34

Quick check


Social Sciences


• Step 3: compute r

Mean 3.6 4.0

2.4-2.6

1.4

-0.6

-0.6

0.0

2.0-2.0

2.0

0.0

-2.0

0.0

6 61 25 6

3 4

3 2

X Y ZX ZY

5.766.76

1.96

0.36

0.36

15.20

€

Y −Y ( )2

4.04.0

4.0

0.0

4.0

16.0Std dev

ZX ZY

1.74 1.790.0

1.1-1.1

0.0

-1.1

1.1

0.0

1.52

X −X( ) X −X( )2

r =ZXZY∑N

Y −Y( )

1.38-1.49

0.8

- 0.34

- 0.34

* =


Social Sciences



Mean 3.6 4.0

2.4-2.6

1.4

-0.6

-0.6

0.0

2.0-2.0

2.0

0.0

-2.0

0.0

6 61 25 6

3 4

3 2

X Y ZX ZY

5.766.76

1.96

0.36

0.36

15.20

€

Y −Y ( )2

4.04.0

4.0

0.0

4.0

16.0Std dev

ZX ZY

1.74 1.790.0

1.1-1.1

0.0

-1.1

1.1

0.0

1.521.64

0.88

0.0

0.37

X −X( ) X −X( )2

r =ZXZY∑N

=4.41

5

Y −Y( )

1.38-1.49

0.8

- 0.34

- 0.34

=0.89

4.41


Social Sciences



Y

X1

2

34

5

6

1 2 3 4 5 6

• Appears linear

• Positive relationship

• Fairly strong relationship• .89 is far from 0, near +1

r =ZXZY∑N

=.89

relationships between variables statistics for the social sciences psychology 340 spring 2010

Documents