Psychology 9

1
Psychology 9

• Quantitative Methods in Psychology
• Jack Wright
• Brown University
• Section 29 (old 31)

Note. These lecture materials are intended
solely for the private use of Brown University
students enrolled in Psychology 9, Spring
Semester, 2002-03. All other uses, including
duplication and redistribution, are unauthorized.

2
Agenda

• Correlation
• Announcements
• final projects see handout
• Schedule
• skip chapter 13 (repeated measures)
• Correlation (Ch. 6) Regression (Ch. 7)
• Chi-Square (Ch. 14)

3
Transition to Correlation

• So far, focused on ONE dependent (aka response)
  variable
• one- and two-sample comparisons
• even paired two-sample t was reduced to ONE
  difference score
• one-way ANOVA
• two-way ANOVA
• In all of these, we focused on shifts in location
• eg whether means differed

4
Transition

• Now turn to a different type of problem
• TWO dependent variables that are paired
• question of interest is not location shift
• but degree to which the variables are associated
• eg height of mothers, heights of daughters
• eg IQ score for brother, IQ score for sister

5
Sir Francis Galton (1822-1911)
6
Next Unrelated children reared together.
7
Next Full siblings children reared together.
8
Next Identical twins reared apart.
9
Next What about other traits (e.g., IQ)?
10
IQ of one twin
Next Identical twins reared apart.
IQ of one twin
11
IQ of one twin
IQ of one twin
12
Goals of our Measure of Association

• 1. assess degree of linear relatedness between
  paired quantitative variables
• 2. express as a pure number that is unaffected by
• shift in location for X vs. Y
• difference in variance between X and Y
• thus, could compare relatedness for heights with
  relatedness for IQ, etc
• 3. make inferences about probability of outcomes
• develop sampling distribution for our statistic

13
Karl Pearson
14
Pearsons Product-Moment Correlation Coefficient

• 1. product based on product of the pairs of
  scores for X and Y
• 2. moment as in first moment or mean,
  signifying mean of the products
• 3. correlation from the original co-relation
  of X and Y
• 4. labelled
• r rho for population parameter
• r sample statistic

15
Understanding the computation of r

• 1. convert X and Y to z-scores Zx, Zy
• removes original means and sets each to 0
• removes original variance and sets each to 1
• 2. form the cross-products of Zx and Zy
• 3. get the mean of the cross-products
• this will be sensitive to degree of relatedness
• example follows

16
Example
X
Y
Zx
Zy
ZxZy
1 3 4 5 7
-1.5 -0.5 0.0 0.5 1.5
5 9 7 1 13
-0.5 0.5 0.0 -1.5 1.5
0.75 -0.25 0.00 -0.75 2.25
Mean
4
7
Szxzy 2.0
s
2
4
Szxzy
2.0
r

.4
N
5
Note N used is pairs and must be consistent.
17
r at work
2
4
1
1
-.25
0
0
IQ of one twin
-.25
-1
1
-2
4
0
1
2
-2
-1
IQ of one twin
18
(No Transcript)
19
(No Transcript)
20
Sampling distributions for r

• 1. assume X,Y pairs, each normally distributed
  N(m,s)
• 2. assume bivariate distribution XY is
  bivariately normal
• peak density is at Mx,My
• slices of Y for each value of X are normal
• slices of X for each value of Y are normal
• true correlation between XY is rho

21
Sampling distributions for r

• 3. take random sample of size N pairs from
  bivariate population
• 4. compute Pearsons r
• 5. accumulate over many samples
• 6. result is the sampling distribution for r
• what is the shape of these sampling
  distributions?
• what is their variability?
• with this information, we can make inferences in
  the usual fashion

22
(No Transcript)
23
(No Transcript)
24
Testing the significance of r

• when r 0 and N is small, the sampling
  distribution of r is
• symmetric around 0
• however, platykurtic rather than normal
• specifically, distributed approximately as
• t with df N - 2

25
Testing the significance of r (for rho0)

• typically use the t distribution to evaluate the
  significance of r

r
tobs
Example r .6 n 10
t .6/ sqrt (1-.36)/8 .6/sqrt.64/8
2.14
t(8) 2.14, p gt .05 (ns).
26
More on testing the significance of r

• when p gt 0 or p lt 0 and N is small, the sampling
  distribution of r is
• skewed rather than symmetric clearly, not
  distributed as t
• solution
• transform r so as to remove skew
• Fishers r-to-z transform
• z 1/2loge(1 r) - loge(1 - r)
• or use r-to-z table
• see handout

27
Effect of Fishers r-to-z transform

• r z
• 0 0
• .2 .20
• .5 .55
• .7 .87 (note pulls to the right)
• .8 1.10 (pulls more)
• .9 1.47
• .99 2.65 (still more)
• Effect stretches values piled up near 1 so that
  right tail is symmetric with left.

28
Testing significance of z

• when r gt 0, r is skewed, but z is approximately
  normally distributed, with standard error

1
sz
So, we can now perform significance tests using
the standard normal distribution
zr - zr
z
sz
29
Testing significance of z

• Example
• r .5, n 10, rho .9

1/sqrt(7) .38
sz
zr .55 zrho 1.47
.92/.38 2.42
z
30
Factors that affect r and its interpretation

• 1. violations of bivariate normality assumption
• 1. non linearities
• 2. heteroscedasticity
• 3. outliers
• remedy
• always examine scatterplots of X Y
• always look for possible violations

31
Factors

• 2. restricted range
• sample is restricted to portion of the range of X
• even when rho is high, sample r will be
  attenuated
• see examples in text

32
Other interpretive problems

• 3. confounds and the problem of interpreting
  causation
• X Y are paired in the world rather than by
  random assignment
• therefore possible confounds with some other
  variable Z are difficult to rule out