What does covariance tell you? What it is a function of? What coefficient tells you the strength of the relationship? What is confidence a function of? - PowerPoint PPT Presentation

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What does covariance tell you? What it is a function of? What coefficient tells you the strength of the relationship? What is confidence a function of?

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... 24616.2231 1097 22.4395835 Root MSE = 4.6898 ----- polpartbes2 | Coef. Std. Err. t P|t| [95% Conf. Interval ... – PowerPoint PPT presentation

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Title: What does covariance tell you? What it is a function of? What coefficient tells you the strength of the relationship? What is confidence a function of?


1
What does covariance tell you?What it is a
function of?What coefficient tells you the
strength of the relationship?What is confidence
a function of?
Review
2
What is the central limit theorem?What is a
normal distribution?What inference does the
central limit theorem help us with?
Review of central limit theorem
3
yi a bxi ei yhat a bx
Two formulas
  • What is yhat?
  • What is yi?
  • What is xi?
  • What is ei?
  • What is b?
  • What is a?

4
Interpreting results
  • What is the difference between b and Beta?
  • What is the standard error
  • How do you compute t?
  • What is the significance level?

5
Residual review
What is a residual? What is the mean of
residuals? What assumption do we make about
residuals?
6
What is a z score?How is it computed?What is
the beta coefficient?How is it different from
the b in terms of interpreting the effect?
Z score review
7
T statistic review
  • What is the formula for the t statistic?
  • If the t 2, how confident are we?
  • (what are we confident about?)

8
The intercept?
  • If the intercept is 3, and the dependent variable
    ranges from 1-4 and the independent variable is
    1-4, what other information do we need to know
    the value of the DV when the IV is at its lowest
    value?
  • The slope is 2.
  • What is the value of the DV when the IV is at its
    lowest value?

9
  • If you multiply the dependent variable by 100,
    what numbers change? How do they change?
  • What numbers do not change?
  • Potential answers
  • B, Beta, standard error, t, significance

10
Where does our estimate of the error come from?
  • The residuals. If the points are far from the
    slope, then we are less confident.
  • If the points are close to the slope, then we are
    more confident.

11
Can we be wrong about rejecting a null
hypothesis?There are two kinds of errors
  • (Type 1) a true null hypothesis can be
    incorrectly rejected
  • (Type 2) a false null hypothesis can fail to be
    rejected.

12
Type 2 error is more serious
  • We you fail to reject the hypothesis, you do not
    prove the hypothesis is wrong. (remember, we
    dont ever prove anything).
  • It could be measurement error and all kinds of
    statistical problems that lead to rejecting a
    null hypothesis.

13
Null Hypothesis Rejected
  • If you reject it, then you have tried to prove
    your theory wrong and you could not.
  • Dont forget that you havent proven anything (we
    never prove anything)
  • You still have other ways of trying to prove it
    wrong

14
What is the question that we ask in statistical
analysis?
  • How much better have we done than the mean in
    predicting values of y from x?

15
How do we know we have done better than the mean?
  • Distance between the slope and the mean is great
  • What is the confidence of doing better than the
    mean likely determined by?
  • Ratio of explained to unexplained variance

16
Wouldnt it be great to have a coefficient that
told us the ratio of explained to unexplained
variance?
  • Total Variance
  • Explained Variance
  • Unexplained variance

17
R square
  • R square Explained Variance
  • Unexplained Explained
  • Variance
  • Unexplained Variance Explained Variance what?
  • (total variance)

18
  • For each observation, you calculate the distance
    from the mean to the slope squared to get
    explained variance.
  • Then divide by the total sum of squares, which is
    total variance

19
Pearson r and R square
  • Pearson r squared is the same as R square
  • (in the bivariate case one independent
    variable)
  • (Pearson r)2 R square
  • R square is standardized and symmetric
  • Symmetric means that it doesnt matter which is
    the independent variable and which is the
    dependent variable

20
Formula for the slope(in the bivariate case)
bx covariation of X and Y Sum(Xi-X)(Yi-Y)
bx variation of X Sum(Xi-X)2
21
Formula for r and beta
Beta is the same as r in the case of bivariate
22
Theory Severity of grievance makes a person more
likely to participate
  • One measure of political participation is a count
    of the number of activities
  • Ranging from 0-31

23
Guttman Scale measure of political participation
How hard the participation is
  • . tab part_category
  • part_category
    Freq. Percent Cum.
  • -------------------------------------------------
    --------------------------
  • Did not participate
    555 50.55 50.55
  • Contacted official in writing or in per
    124 11.29 61.84
  • Participated in rally
    274 24.95 86.79
  • Participated in illegal activity or hun
    145 13.21 100.00
  • -------------------------------------------------
    --------------------------
  • Total
    1,098 100.00

24
The effect on the number of family members that
were victims on count
. regr polpartbes2 relatives_victims
Source SS df MS
Number of obs 1098 -------------------------
------------------ F( 1, 1096)
23.19 Model 510.137776 1
510.137776 Prob gt F 0.0000
Residual 24106.0854 1096 21.9946034
R-squared 0.0207 ------------------------
------------------- Adj R-squared
0.0198 Total 24616.2231 1097
22.4395835 Root MSE
4.6898 ------------------------------------------
------------------------------------ polpartbes2
Coef. Std. Err. t Pgtt 95
Conf. Interval ---------------------------------
-------------------------------------------- relat
ives_s -.5865734 .121797 -4.82 0.000
-.8255551 -.3475917 _cons 3.486299
.2412254 14.45 0.000 3.012983
3.959614 -----------------------------------------
-------------------------------------
25
The effect on the number of family members that
were victims on Guttman scale
. regr part_category relatives_victims
Source SS df MS
Number of obs 1098 -------------------------
------------------ F( 1, 1096)
5.42 Model 6.93949761 1 6.93949761
Prob gt F 0.0200 Residual
1401.98673 1096 1.27918497 R-squared
0.0049 ------------------------------------
------- Adj R-squared 0.0040
Total 1408.92623 1097 1.28434479
Root MSE 1.131 -------------------------
--------------------------------------------------
--- part_categy Coef. Std. Err. t
Pgtt 95 Conf. Interval ----------------
--------------------------------------------------
----------- relatives_s -.0684136 .0293728
-2.33 0.020 -.1260469 -.0107804
_cons 1.11792 .0581744 19.22 0.000
1.003775 1.232066 ----------------------------
--------------------------------------------------

26
The effect on the number of family members that
died on count
. regr polpartbes2 relatives_died
Source SS df MS
Number of obs 1098 -------------------------
------------------ F( 1, 1096)
9.00 Model 200.387171 1 200.387171
Prob gt F 0.0028 Residual
24415.836 1096 22.2772226 R-squared
0.0081 ------------------------------------
------- Adj R-squared 0.0072
Total 24616.2231 1097 22.4395835
Root MSE 4.7199 -------------------------
--------------------------------------------------
--- polpartbes2 Coef. Std. Err. t
Pgtt 95 Conf. Interval ----------------
--------------------------------------------------
----------- relatives_d .4331152 .1444106
3.00 0.003 .1497628 .7164676
_cons 2.248116 .1735599 12.95 0.000
1.907569 2.588663 ----------------------------
--------------------------------------------------

27
The effect on the number of family members that
died on Guttman scale
. regr part_category relatives_died
Source SS df MS
Number of obs 1098 -------------------------
------------------ F( 1, 1096)
1.96 Model 2.51377401 1 2.51377401
Prob gt F 0.1619 Residual
1406.41246 1096 1.28322304 R-squared
0.0018 ------------------------------------
------- Adj R-squared 0.0009
Total 1408.92623 1097 1.28434479
Root MSE 1.1328 -------------------------
--------------------------------------------------
--- part_categy Coef. Std. Err. t
Pgtt 95 Conf. Interval ----------------
--------------------------------------------------
----------- relatives_d .04851 .0346593
1.40 0.162 -.019496 .1165161
_cons .9748847 .0416553 23.40 0.000
.8931516 1.056618 ----------------------------
--------------------------------------------------

28
The effect on the severity of psychological
damage on respondent on count
. regr polpartbes2 sevpsych Source
SS df MS Number of
obs 316 -----------------------------------
-------- F( 1, 314) 9.47
Model 210.190105 1 210.190105
Prob gt F 0.0023 Residual
6967.51876 314 22.1895502 R-squared
0.0293 ------------------------------------
------- Adj R-squared 0.0262
Total 7177.70886 315 22.7863773
Root MSE 4.7106 -------------------------
--------------------------------------------------
--- polpartbes2 Coef. Std. Err. t
Pgtt 95 Conf. Interval ----------------
--------------------------------------------------
----------- sevpsych .9232447 .2999749
3.08 0.002 .3330298 1.51346
_cons -.7109326 1.124305 -0.63 0.528
-2.923056 1.50119 ----------------------------
--------------------------------------------------

29
The effect on the severity of psychological
damage on respondent on Guttman scale
. regr part_category sevpsych Source
SS df MS Number of
obs 316 -----------------------------------
-------- F( 1, 314) 4.29
Model 5.28062712 1 5.28062712
Prob gt F 0.0392 Residual
386.678234 314 1.23145934 R-squared
0.0135 ------------------------------------
------- Adj R-squared 0.0103
Total 391.958861 315 1.24431384
Root MSE 1.1097 -------------------------
--------------------------------------------------
--- part_categy Coef. Std. Err. t
Pgtt 95 Conf. Interval ----------------
--------------------------------------------------
----------- sevpsych .1463368 .0706677
2.07 0.039 .0072948 .2853788
_cons .5650835 .2648621 2.13 0.034
.0439547 1.086212 ----------------------------
--------------------------------------------------
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