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?
1What 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
2What 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
3yi 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?
4Interpreting results
- What is the difference between b and Beta?
- What is the standard error
- How do you compute t?
- What is the significance level?
5Residual review
What is a residual? What is the mean of
residuals? What assumption do we make about
residuals?
6What 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
7T statistic review
- What is the formula for the t statistic?
- If the t 2, how confident are we?
- (what are we confident about?)
8The 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
10Where 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.
11Can 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.
12Type 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.
13Null 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
14What 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?
15How 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
16Wouldnt it be great to have a coefficient that
told us the ratio of explained to unexplained
variance?
- Total Variance
-
- Explained Variance
-
- Unexplained variance
17R 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
19Pearson 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
20Formula 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
21Formula for r and beta
Beta is the same as r in the case of bivariate
22Theory 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
23Guttman 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
24The 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 -----------------------------------------
-------------------------------------
25The 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 ----------------------------
--------------------------------------------------
26The 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 ----------------------------
--------------------------------------------------
27The 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 ----------------------------
--------------------------------------------------
28The 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 ----------------------------
--------------------------------------------------
29The 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 ----------------------------
--------------------------------------------------