These are the notes of the lectures after the midterm, i.e. lectures 9 to 14, for the course Statistics given at Utrecht University with course code GEO2-2217 for GSS and NW&I students. (see also my other summary for part 1).
Part 2: Statistics
Lecture 9: Association interval and ordinal variables 2
Association interval and ordinal variables 2
Covariance and Pearson’s r 3
Rank correlation (rS and tau) 5
Correlation tests 6
Partial correlation 7
Lecture 10: Linear Regression part 1 8
Linear regression in case of one independent variable (simple regression) 8
Significance b 10
Predictive use 12
Multiple regression: linear regression in case of multiple independent variables 13
Determine influence difference variables: 14
Part (semi-partial) and partial correlation 14
Significance 15
R^2 16
Testing H0: and confidence intervals ’s 16
Building a regression model 17
Dummy variable 17
Including interaction terms 19
Steps in multiple regression: 20
Lecture 11: Linear Regression part II 20
Missing values 21
Assessing the model & checking assumptions 21
Outliers: 21
Assumptions: 22
1. Independence of observations (each observation ones in your sample) 22
2. Non-zero variance (we need variation in the data, not all data is the same) 22
3. Type and distribution variables: 22
4. No Multicollinearity 22
5. Homoscedasticity 23
6. Normally distributed errors 23
7. Linearity (not curved) 23
8. Bundles of parallel regression lines 25
9. Independent errors (you can check this via Durbin Watson test, not incl. in the
course) 25
Path Analysis and Mediation effects 26
Lecture 12: Association nominal variables 31
Association significant? Chi-square goodness-of-fit test 31
From Phi to Carmer’s V 34
Special measures for 2x2 tables 34
SPSS 35
1
, Moderator: Role variable W 36
Common ORmh (Mantel Haenszel) 38
Steps cross table analysis 38
Association measures nominal variables 39
Lecture 13: Logistic Regression 39
Linear regression error with dichotomous Y 39
From dichotomous to continuous Y 40
Conversion formulas 40
ML-estimates of the 𝛽s 41
Interpretation of the 𝛽s 41
Significance of an effect (Wald test or CI around the OR) 42
Assessing the model 43
Lecture 14: factor analysis 48
Factor score and factor loading 49
Summary in between: factor analysis 50
Idea factor analysis 51
Reproduced and residual correlations 52
Interpretation 53
Evaluation of the scale 54
When can you do factor analysis? 54
Extra aantekeningen 54
Degrees of freedom / effect sizes / difference Z and T 54
Chi-square tests: for categorical tests 55
2
,Lecture 9: Association interval and ordinal
variables
Practical 7: Association interval and ordinal variables and checking
an association for a third variable
Association interval and ordinal variables
Explanation: if there are two variables and a high value for Y is more common with also a
high value of X, we call this cohesive, associative or correlative variable.
Negative correlation = higher values for one variable go together with low values for the
other variable
Strongest correlation in a plot is when it is closed to a descending regression line and
weakest when there is barely a rise or fall noticeable
Measurement level Trend Measures (Greek
letter for Population)
X Y
scale scale 1. linear 1. Pearson’s r (p)
2. curve 2. Spearman’s
(ps) Kendall's
Tau (t)
ordin ordin increasing/decreasing Spearman’s rs(ps),
Kendall’s tau (t)
scale nomin increasing/decreasing Eta^2 (n^2)
nomin nomin - Phi (p), Carmer’s V
dich dich - Phi(P), Odds ratio OR
N.B. : r, rS , tau = -1<>+1
Calculate center of gravity coordinate = (all x values divided by number of x values) (all y
values divided by number of y values)
Covariance and Pearson’s r
Covariance shows you how the two variables differ, whereas correlation shows you how the
two variables are related.
Covariance can be positive or negative ; it is an indication for correlation, and is the
“combined variance”.
Calculating the covariance =
3
, - First you calculate the deviation of every point from the center of graffiti both x and y
times each other
- than you sum all these deviations up
- than you divide this by the number of observations minus 1
Pearson’s R
- To determine the strength of the correlation, there is a correlation measure called
Pearson’s r or the product moment correlation coefficient.
The R value is equal to the standardized covariance, which arise from the deviations by
dividing the standard deviation of the variable
Covariance and correlation coefficient (r) are related measures of the linear relationship
between two variables. The covariance (cov) measures how much two variables vary
together, while the correlation coefficient (r) measures the strength and direction of the linear
relationship between two variables.
r = COVxy / (Sx*Sy)
r is not scale-sensitive but Covariance is!!
Interpretation of R=
- r = coefficient of linear association (standardized covariance)
- -1 <r < +1; sign r: positive / negative correlation
- r = standardized regression coefficient b in case of simple regression
- r^2 = proportion variation in y linearly explained by X
- covariance is not an association measure
r^2 <0.9: weak linear association
.09 < r2 < 0.25 = medium linear association
r^2 >0.25 = strong linear association
Example:
- if r = -.5
- clearly negative correlation
- a 1.0 sx increase in x associates with a 0.5 sy decrease in y
- r^2 -.25 : 25% Y-variation linearly explained by X
Eta versus r
Eta:
- more general measure for dependency Y on X
- eta^2 = proportion variation Y explained by x
4
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