Bivariate Relationships

1
Bivariate Relationships

• Standardization and Confidence

2
Standardized variable review

• Z scores are linear transformations of variables
• Z score (x) (x-mean of x) /standard deviation
  of x

3
Z scores

• Z scores always have
• a mean of zero
• a standard deviation of 1

4
Histogram of Happiness
60
50
40
Percent
30
20
10
0
Not Too Happy
Pretty Happy
Very Happy
1
2
3
General Happiness
5
Histogram of z score of happiness
60
50
40
Percent
30
20
10
0
Not Too Happy
Pretty Happy
Very Happy
-1.3
.12
1.6
General Happiness
6
Descriptives Syntax
desc vars happy zhappy.
7
Descriptives of Happiness and z score of happiness
8
What is the correlation between zhappy and happy?
9
Normal distribution review
10

• Approximately 68 percent of the area under a
  normal curve lies between the values of the mean
  and the standard deviation and the mean.

11

• Approximately 95 of the area lies between 2
  standard deviations and the mean.

12

• Approximately 99.7 lies between 3 standard
  deviations and the mean.

13
What is the mean of a z score?
14
What is the null hypothesis?

• What are we trying to reject in a null hypothesis?

15
Not Zero

• So, we are more confident as we believe that the
  slope is not zero.
• We know that the area under the normal curve at 2
  standard deviations away from zero (the mean) is
  2.5 of the area of the curve (approximately).
• We also know that 2 standard deviations away from
  the mean in the other direction is 2.5 of the
  area of the curve.

16
T statistic
95
2.5
2.5
0
mean
If the slope falls out of the range of two
standard deviations from 0 then we can say that
we are 95 confident that the relationship is not
zero.
17
Formula for t

• T slope/standard error
• If the t is at least 2, then it is two standard
  deviations from the mean of the curve which is
  zero (why is it 0?), then we are 95 confident
• Significance is a linear transformation of the t
  statistic based on the theory of the normal
  curve.
• Also known as p values.

18
How confident are we?
If the slope falls within two standard deviations
from zero, then we have a difficult time saying
that we are confident. Since we can say with
precision what the probability is that the
relationship from the population could be zero,
then we know how confident we are.
19

• Approximately 68 percent of the area under a
  normal curve lies between the values of the mean
  and the standard deviation and the mean.

If t 1, then we are 68 confident. That is not
very confident.
20
Approximately 99.7 lies between 3 standard
deviations and the mean. If the t 3, then
we are 99.7 confident.
21
One tailed versus two tailed test
95
2.5
2.5
You can use theory to rule out one of the areas
covering 2.5. If you know the slope should be
positive, then you can cross out the 2.5 on the
left. Then you are 97.5 confident that the
relationship is not zero.
22
One tailed versus two tailed test
95
2.5
2.5
You can use theory to rule out one of the areas
covering 2.5. If you know the slope should be
negative, then you can cross out the 2.5 on the
left. Then you are 97.5 confident that the
relationship is not zero.
23
Covariation

• When it tends to be the case that x is greater
  than the mean when y is greater than the mean AND
  x is lower than the mean when y is lower than the
  mean, then there is a positive covariation

24
Plot showing positive covariance
Mean urban
Mean female literacy
25
Expected value

• But we may want to know more specific knowledge
  than that we may want to know the expected
  value of y for each increased value of x
• I may know the mean of everyones height in class
• But if I know gender, then I can generate two
  expected values
• If you remember, we are always trying to do
  better than the mean

26
Regression analysisimportant to know
substantive effect

• For every 10K dollars given in humanitarian aid,
  there is an increase in 3K spent on weapons
• Different from every 10K dollars given in
  humanitarian aid, there is a .5K increase spent
  on weapons
• Different from every 10K dollars given in
  humanitarian aid, there is a 8K increase spent on
  weapons
• Unit of analysis?

27
Life Happiness and Occupational Prestige
28
Life Happiness and Prestige
3
2.5
Happiness
2
1.5
R

S
q

L
i
n
e
a
r



0
.
2
6
8
1
20
30
40
50
R's Occupational Prestige Score (1980)
29
Regression equation

• Y a bx e
• y a bx
• y is also known as yhat
• y is the dependent variable value
• yhat is the predicted value
• a is the intercept

30
Regression Syntax

• Syntax is regr DV IV
• regr happy prest80, beta
• Reports beta coefficients same as Pearson r
  (when there is only one independent variable)
• regr happy prest80
• Reports confidence intervals instead of betas

31
Regression results
Source SS df MS
Number of obs 11 -------------------------
------------------ F( 1, 9)
3.30 Model 1.31753739 1 1.31753739
Prob F 0.1026 Residual
3.59155351 9 .399061502 R-squared
0.2684 ------------------------------------
------- Adj R-squared 0.1871
Total 4.90909091 10 .490909091
Root MSE .63171 -------------------------
--------------------------------------------------
--- happy Coef. Std. Err. t
Pt Beta ----------------
--------------------------------------------------
----------- prestg80 -.0380391 .0209348
-1.82 0.103 -.518061
_cons 3.330371 .8050567 4.14 0.003
. ----------------------------
--------------------------------------------------
Why are we not that confident in our results? Why
is the beta so much larger than the coefficient
for the slope?
32
Regression results
Source SS df MS
Number of obs 11 -------------------------
------------------ F( 1, 9)
3.30 Model 244.378788 1 244.378788
Prob F 0.1026 Residual
666.166667 9 74.0185185 R-squared
0.2684 ------------------------------------
------- Adj R-squared 0.1871
Total 910.545455 10 91.0545455
Root MSE 8.6034 -------------------------
--------------------------------------------------
--- prestg80 Coef. Std. Err. t
Pt Beta ----------------
--------------------------------------------------
----------- happy -7.055556 3.88302
-1.82 0.103 -.518061
_cons 50.83333 7.853795 6.47 0.000
. ----------------------------
--------------------------------------------------
Why is the coefficient so much bigger? What
happens to the confidence? Why is the beta the
same?
33
(No Transcript)
34
What happens to the confidence if we keep the
slope the same but double the n?
35
What happens to the confidence if we keep the
doubled n but decrease the variance of
occupational prestige?
36
Syntax

• if prestg80
• if prestg80 32 occpres 2.
• if prestg80 45 occpres 3.
• execute.
• add value labels occpres 1 "Not that Prestigious
• 2 "Pretty
  Prestigious"
• 3 "Very Prestigious".

37
. sum prestg80 Variable Obs
Mean Std. Dev. Min
Max ---------------------------------------------
------------------------ prestg80 11
37.36364 9.542251 22 51
38
(No Transcript)
39
(No Transcript)
40
Regression results
Source SS df MS
Number of obs 11 -------------------------
------------------ F( 1, 9)
3.30 Model 244.378788 1 244.378788
Prob F 0.1026 Residual
666.166667 9 74.0185185 R-squared
0.2684 ------------------------------------
------- Adj R-squared 0.1871
Total 910.545455 10 91.0545455
Root MSE 8.6034 -------------------------
--------------------------------------------------
--- prestg80 Coef. Std. Err. t
Pt 95 Conf. Interval ----------------
--------------------------------------------------
----------- happy -7.055556 3.88302
-1.82 0.103 -15.83956 1.728447
_cons 50.83333 7.853795 6.47 0.000
33.06681 68.59985 ----------------------------
--------------------------------------------------
Why are we not that confident in our results?
41
Regression results
Source SS df MS
Number of obs 11 -------------------------
------------------ F( 1, 9)
3.30 Model 244.378788 1 244.378788
Prob F 0.1026 Residual
666.166667 9 74.0185185 R-squared
0.2684 ------------------------------------
------- Adj R-squared 0.1871
Total 910.545455 10 91.0545455
Root MSE 8.6034 -------------------------
--------------------------------------------------
--- prestg80 Coef. Std. Err. t
Pt 95 Conf. Interval ----------------
--------------------------------------------------
----------- happy -7.055556 3.88302
-1.82 0.103 -15.83956 1.728447
_cons 50.83333 7.853795 6.47 0.000
33.06681 68.59985 ----------------------------
--------------------------------------------------
Why are we not that confident in our results?
42
Regression results
Source SS df MS
Number of obs 11 -------------------------
------------------ F( 1, 9)
3.30 Model 244.378788 1 244.378788
Prob F 0.1026 Residual
666.166667 9 74.0185185 R-squared
0.2684 ------------------------------------
------- Adj R-squared 0.1871
Total 910.545455 10 91.0545455
Root MSE 8.6034 -------------------------
--------------------------------------------------
--- prestg80 Coef. Std. Err. t
Pt 95 Conf. Interval ----------------
--------------------------------------------------
----------- happy -7.055556 3.88302
-1.82 0.103 -15.83956 1.728447
_cons 50.83333 7.853795 6.47 0.000
33.06681 68.59985 ----------------------------
--------------------------------------------------
Why are we not that confident in our results?