Alcohol consumption and HDI story

Total Beer Wine Spirits Other HDI Lifetime span

Austria 13,24 6,7 4,1 1,6 0,4 0,755 80,119

Finland 12,52 4,59 2,24 2,82 0,31 0,800 79,724

Poland 13,25 4,72 3,26 1,56 0 0,715 75,976

Russia 15,76 3,65 0,1 6,88 0,34 0,644 67,260

Uganda 11,93 0,51 0 0,18 14,52 0,453 53,261

The Human Development Index (HDI) is a composite statistic of life expectancy, education, and income

What is a CORRELATION

Correlation – statistical procedure to measure & describe the relationship between two variable

Do two variables covary?

Are two variables dependent or independent of one another?

Can one variable be predicted from another?

What is a CORRELATION

World is full of COVARY

The IQ and brain sizeIQ Z IQ Pixel countZ PC CP

138 1,2947 991 2,122 2,74793140 1,37264 856 0,046 0,06333

96 -0,3421 879 0,4 -0,1367983 -0,8487 865 0,185 -0,15664

101 -0,1472 808 -0,69 0,10189135 1,17779 791 -0,95 -1,1231

85 -0,7708 799 -0,83 0,6401377 -1,0825 794 -0,91 0,9823188 -0,6538 894 0,631 -0,4123

Mean 104,78 853 2,70678SD 25,66 65,0192

n= 9 r= 0,33835r= 0,33835

Pearson's product-moment coefficient

.0 to .2 No relationship to very weak association.2 to .4 Weak association.4 to .6 Moderate association.6 to .8 Strong association.8 to 1.0 Very strong to perfect association

Interpretation

CAUTION!!!

Test the null

Testing H0

𝑡=𝑟 √ 𝑛−21−𝑟 2

Alcohol consumption and HDI story

total beer wine spirits other HDI Lifetimetotal 1,00 0,76 0,60 0,61 0,18 0,55 0,34beer 0,76 1,00 0,46 0,44 -0,13 0,63 0,46wine 0,60 0,46 1,00 0,16 -0,12 0,51 0,46spirits 0,61 0,44 0,16 1,00 -0,17 0,48 0,38other 0,18 -0,13 -0,12 -0,17 1,00 -0,25 -0,37HDI 0,55 0,63 0,51 0,48 -0,25 1,00 0,84Lifetime 0,34 0,46 0,46 0,38 -0,37 0,84 1,00

Correlation and causation

B causes A (reverse causation)The more firemen fighting a fire, the bigger the fire is observed to be.Therefore firemen cause an increase in the size of a fire.

A causes B and B causes A (bidirectional causation)Increased pressure is associated with increased temperature.Therefore pressure causes temperature.

Third factor C (the common-causal variable) causes both A and B)Sleeping with one's shoes on is strongly correlated with waking up with a headache.Therefore, sleeping with one's shoes on causes headache.

Illogically inferring causation from correlation

CoincidenceWith a decrease in the wearing of hats, there has been an increase in global warming over the same period.Therefore, global warming is caused by people abandoning the practice of wearing hats.

 

Church of the Flying Spaghetti Monster

Alcohol consumption and HDI story

total beer wine spirits other HDI Lifetimetotal 1,00 0,76 0,60 0,61 0,18 0,55 0,34beer 0,76 1,00 0,46 0,44 -0,13 0,63 0,46wine 0,60 0,46 1,00 0,16 -0,12 0,51 0,46spirits 0,61 0,44 0,16 1,00 -0,17 0,48 0,38other 0,18 -0,13 -0,12 -0,17 1,00 -0,25 -0,37HDI 0,55 0,63 0,51 0,48 -0,25 1,00 0,84Lifetime 0,34 0,46 0,46 0,38 -0,37 0,84 1,00

ScatterplotScatter plot of spousal ages, r = 0.97

Scatter plot of Grip Strength and Arm Strength, r = 0.63

Farnsworth favorite game

Anscombe’s quartetI II III IV

x y x y x y x y

10.0 8.04 10.0 9.14 10.0 7.46 8.0 6.58

8.0 6.95 8.0 8.14 8.0 6.77 8.0 5.76

13.0 7.58 13.0 8.74 13.012.74

8.0 7.71

9.0 8.81 9.0 8.77 9.0 7.11 8.0 8.84

11.0 8.33 11.0 9.26 11.0 7.81 8.0 8.47

14.0 9.96 14.0 8.10 14.0 8.84 8.0 7.04

6.0 7.24 6.0 6.13 6.0 6.08 8.0 5.25

4.0 4.26 4.0 3.10 4.0 5.39 19.012.50

12.010.84

12.0 9.13 12.0 8.15 8.0 5.56

7.0 4.82 7.0 7.26 7.0 6.42 8.0 7.91

5.0 5.68 5.0 4.74 5.0 5.73 8.0 6.89

Property Value

Mean of x in each case 9

Variance of x in each case 11

Mean of y in each case 7.50 

Variance of y in each case

4.122 or 4.127

Correlation between x and y in each case

0.816 

Anscombe’s quartetI II III IV

x y x y x y x y

10.0 8.04 10.0 9.14 10.0 7.46 8.0 6.58

8.0 6.95 8.0 8.14 8.0 6.77 8.0 5.76

13.0 7.58 13.0 8.74 13.012.74

8.0 7.71

9.0 8.81 9.0 8.77 9.0 7.11 8.0 8.84

11.0 8.33 11.0 9.26 11.0 7.81 8.0 8.47

14.0 9.96 14.0 8.10 14.0 8.84 8.0 7.04

6.0 7.24 6.0 6.13 6.0 6.08 8.0 5.25

4.0 4.26 4.0 3.10 4.0 5.39 19.012.50

12.010.84

12.0 9.13 12.0 8.15 8.0 5.56

7.0 4.82 7.0 7.26 7.0 6.42 8.0 7.91

5.0 5.68 5.0 4.74 5.0 5.73 8.0 6.89

Property Value

Mean of x in each case 9

Variance of x in each case 11

Mean of y in each case 7.50 

Variance of y in each case

4.122 or 4.127

Correlation between x and y in each case

0.816 

CAUTION!!!

Check scatterplot

Anscombe’s quartet

Problems

AGE

8070605040302010

SE

10

8

6

4

2

0

AGE

40302010

SE

9

8

7

6

5

4

3

2

1

Problems: Outliers

r=0,63 r=0,23

Problems: Range restriction

Coefficient of Determination (r2)

CoD = The proportion of variance or change in one variable that can be accounted for by another variable.

Problems: Range restriction

REGRESSION MODELS

Multiple linear regression (MLR) is a multivariate statistical technique for examining the linear correlations between two or more independent variables (IVs) and a single dependent variable (DV).

MLR

MLR

Poverty prediction

Poverty prediction

Name of regionPopulation change in 10 years.No. of persons employed in agriculturePercent of families below poverty levelResidential and farm property tax ratePercent residences with telephonesPercent rural populationMedian ageNumber of African/Americans

Level of measurementIVs: MLR involves two or more continuous (interval or ratio) or nominal variables (require recoding into dummy variables)DV: One continuous (interval or ratio) variable

Sample sizeTotal N based on ratio of cases to IVs:

Min. 5 cases per predictor (5:1)Ideally 20 cases per predictor (20:1)

LinearityAre the bivariate relationships linear?Check scatterplots and correlations between the DV (Y) and each of the IVs (Xs)Check for influence of bivariate outlier

MulticollinearityIs there multicollinearity between the IVs? (i.e., are they

overly correlated e.g., above .7?)Homoscedasticity

The variance of the error is constant across observations.Check scatterplots between Y and each of Xs and/or check scatterplot of the residuals (ZRESID) and predicted values (ZPRED)

MLR: Pre-analysis assumptions

MLR: Dummy coding for nominal data

Scatterplot: POP_CHNG vs. PT_POOR (Casewise MD deletion)

PT_POOR = 26,186 - ,4037 * POP_CHNG

Correlation: r = -,6491

-20 -10 0 10 20 30 40 50

POP_CHNG

10

15

20

25

30

35

40

45

PT

_PO

OR

95% confidence

MLR: Main Idea

Scatterplot: POP_CHNG vs. PT_POOR (Casewise MD deletion)

PT_POOR = 26,186 - ,4037 * POP_CHNG

Correlation: r = -,6491

-20 -10 0 10 20 30 40 50

POP_CHNG

10

15

20

25

30

35

40

45

PT

_PO

OR

95% confidence

MLR: Main Idea

Poverty prediction

3D Surface Plot of PT_POOR against POP_CHNG and PT_RURAL

POVERTY.STA 8v*30c

PT_POOR = 16,6681-0,3979*x+0,1339*y

> 40 < 40 < 30 < 20 < 10 < 0

-20-10

010

2030

4050

POP_CHNG

020

4060

80100

120

PT_RURAL

0

5

10

15

20

25

30

35

40

45

PT_POOR

Poverty prediction

MLR: Post-analysis assumptions

Multivariate outliersCheck whether there are influential multivariate outlying cases using Mahalanobis' Distance (MD) & Cook’s D (CD).

Normality of residualsResiduals are more likely to be normally distributed if each of the variables normally distributedCheck histograms of all variables in an analysisNormally distributed variables will enhance the MLR solution

Scatterplot: POP_CHNG vs. PT_POOR (Casewise MD deletion)

PT_POOR = 26,186 - ,4037 * POP_CHNG

Correlation: r = -,6491

-20 -10 0 10 20 30 40 50

POP_CHNG

10

15

20

25

30

35

40

45

PT

_PO

OR

95% confidence

MLR: Post-analysis assumptions

Distribution of Raw residuals

Expected Normal

-8 -6 -4 -2 0 2 4 60

1

2

3

4

5

6

7

8

9

10

No

of o

bs

Raw residuals vs. PT_RURAL

Raw residuals = -,2E-6 + 0,0000 * PT_RURAL

Correlation: r = ,58E-7

0 20 40 60 80 100 120

PT_RURAL

-6

-4

-2

0

2

4

6

Raw

res

idua

ls95% confidence

Poverty prediction

MLR: Types of MLR

Direct (or Standard) •All IVs are entered simultaneously

Hierarchical•IVs are entered in steps, i.e., some before others•Interpret R2 change

Forward •The software enters IVs one by one until there are no more significant IVs to be entered

Backward •The software removes IVs one to one until there are no more non-significant IVs to removed

Stepwise •A combination of Forward and Backward MLR

MLR: TOTAL

1. Conceptualise the model 2. Recode predictors (if necessary)3. Check assumptions4. Choose the type of MLR5. Interpret statistical output and meaning of results. 6. Depict the relationships in a path diagram or Venn

diagram 7. Regression equation: If relevant and useful, interpret Y-

intercept and write a regression equation for predicting Y