1. Lecture 4 Statistical analysis
Heidi Hogset
September, 2014

2. Lecture aim and objectives
Investigate methods of statistical analysis
Objectives
Research questions and hypotheses
Statistical tests
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3. Research questions Survey research is all about asking questions
Descriptive questions, univariate – covered in lecture 3
E.g., how has gender composition of enrolled students changed over the last 20 years?
Causal relationships, multivariate – topic today
E.g., what is the relationship between unemployment rates and applications for admission to higher education?
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4. Research questions The research question is The variables required are
Does the unemployment rate influence people’s propensity to seek higher education?
The variables required are
Unemployment rate each year in Norway (choose period)
Number of applications for admission submitted to colleges and universities in Norway each year (same period)
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5. Research questions Suggested causal relationship?
Source for left image: MS clipart
Source for right image:
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6. Research questions
Hypothesis – a statement to test a particular proposition
Example:
In periods with higher levels of unemployment, colleges and universities receive more applications for admission
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7. Research questions Observed trends Variables:
Share of population in each age bracket that is enrolled in higher education (%)
Share of labor force in age bracket that is registered as totally unemployed (%)
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8. Research questions
We assume causal relationship goes from unemployment to school applications, not vice versa
“Applications” is a Dependent variable (DV)
“Unemployment” is an Independent variable (IV)
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9. Research questions Null hypothesis
There is NO relationship between unemployment rates and school applications
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10. Research questions Alternative hypotheses
There is a significant relationship between unemployment rates and school applications (non-directional)
Two-tailed
There is a significant and positive relationship between unemployment and school applications (directional)
One-tailed
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11. Research questions Hypothesis testing One-tailed Two-tailed
Use 2-tailed unless you have a good reason to choose 1-tailed
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12. Your survey
You are expected to develop (a) research question(s) based on theory developed in your discipline of interest
Example: Green Taxing
Does it work?
How strong is the effect?
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13. Your survey Variables needed?
How might you create the variables using a survey?
What hypothesis might you use?
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14. Statistical analysis
The significance of each hypothesis is tested using statistical analysis
The objective is to reject or accept each hypothesis
“Accept” means you have not disproved it
“Accept” does not mean the hypothesis has been proved
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15. Statistical analysis Relationship between two variables
One-Sample T Test
Paired Samples T Test
Independent Samples T Test
Chi-square Test
One-Way ANOVA
Correlation analysis
Simple Regression Analysis
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16. Statistical analysis Relationship between > two variables
Multiple Regression Analysis
Logistic Regression Analysis
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17. Compare two variables – example 1
Number of students aged enrolled in higher education in Norway, by sex,
“Male” is number of male students
“Female” is number of female students
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
Male
27 826
27 103
28 040
28 151
28 612
30 817
30 099
31 478
37 564
40 345
42 544
Female
31 721
31 827
33 669
33 204
33 392
37 062
37 122
39 690
46 188
49 131
51 446
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18. One Sample T Test Compute difference: Female – Male
Test: Is the difference significantly different from zero?
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19. One Sample T Test
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20. One Sample T Test
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21. One Sample T Test
If our test is directional, we should use the one-tailed significance, which is half of the 2-tailed significance. However, here, 0,000/2 = 0,000: No difference.
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22. Paired Samples T Test Compare variables directly: Female vs. Male
Test: Are their means significantly different?
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23. Paired Samples T Test
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24. Paired Samples T Test
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25. Paired Samples T Test
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26. Independent Samples T Test
Has the finance crisis in influenced enrolment in higher education?
Test: Are the means significantly different before and after 2007?
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27. Independent Samples T Test
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28. Independent Samples T Test
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29. Independent Samples T Test
Note – the difference between the means is not only due to the finance crisis. There is also a long-term trend towards higher enrolment rates that we should have controlled for.
The F is Levene’s test for Equality of Variances. A small value of significance here indicates that the appropriate T Test is that where Equal variances are not assumed.
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30. Compare two variables – example 2
A survey of pigeonpea farmers in Tanzania in 2008 – variables:
District (there are 4)
Respondent characteristics (there are 609 respondents)
The respondent’s sex, age, number of years in school, number of dependents in the household, distance to the nearest main market (for farm products)
The respondent’s farm operation
The number of plots planted in improved varieties of pigeonpeas divided by the total number of plots planted in pigeonpeas (“share”)
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31. Chi-square Test
We need to compare two nominal or ordinal variables (discrete data)
We want to check if our sampling procedure has produced a biased sample with respect to gender composition
We assume the proportion of households with a female household head is independent of districts
Test: Is the sex distribution different between districts?
Chi-square Test for Independence of Discrete Data
Data for which the only meaningful statistics are frequencies and percentages
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32. Chi-square Test
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33. Chi-square Test Select Statistics, then tick Chi-square
Select Cells, then tick Expected
(and untick Observed)
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34. Chi-square Test Chi-square Test of Independence
Karatu o
Arumeru o
Chi-square Test of Independence
A small chi-square statistic indicates that there is a significant relationship between the two variables – they are NOT independent of each other
Here, we have a large number, close to 1. Therefore, we ACCEPT the null that the two variables are independent
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35. One-Way ANOVA Compare two variables Procedure:
Test 1: Is the share of pigeonpea fields in improved varieties different between districts?
Test 2: Does the share of pigeonpea fields in improved varieties vary by the farmer’s school experience?
Procedure:
Analyze/ Compare Means/ One-Way ANOVA/ Select DV (Share improved pigeonpeas)/ Select factor (District)/ OK
The DV should be interval or ratio data type
The populations should be normally distributed and the population variances should be equal.
This procedure becomes cumbersome when the number of factors goes beyond 3-4.
Mainly used in psychological research using experimental data, which is not common in economics research
Problem: Is Share an interval or ratio type data?
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36. One-Way ANOVA
The variable Share has range from 0 (no improved pigeonpea) to 1 (only imporved pigeonpea), with values clustering around the values 0 (75,5% of observations) and 1 (15,3%), ¼, ¾, ½, 1/3, 2/3
The variable School measures the number of years the respondent has attended school. Observations vary from 0 to 16.
There are 4 districts.
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37. Correlation analysis
Examines bivariate relationships between ≥ 2 ORDINAL or INTERVAL/ RATIO variables
They are CORRELATED if they are systematically related
Positively: The variables tend to move in the same direction
Negatively: The variables tend to move in opposite directions
Un-correlated: No relationship
Can be run with any kind of data, but is not appropriate for nominal variables with more than 2 categories
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38. Correlation analysis
Correlation is measured by the correlation coefficient, r
Helps to think of correlation in visual terms
Perfect –
Mod. –
No rel.
Mod. +
Perfect + -1
-0.7
-0.5
-0.1
0.1
0.5
0.7 1
Strong –
Weak –
Weak +
Strong +
Mod. = Moderate
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39. Correlation analysis
r  -1
r  1
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40. Correlation analysis
r  0?
r  0
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41. Correlation analysis Scatter-plot procedure in SPSS Graphs
Legacy Dialogs
Scatter/ Dot
Select Simple Scatter
Define
IV for x-axis
DV for y-axis OK
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42. Correlation analysis Correlation procedure SPSS Analyze Correlate
Bivariate
Add variables to variables list
Tick Pearson’s for interval/ ratio data (Spearman’s for ordinal) OK
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43. Correlation analysis
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44. Correlation analysis
Correlation shows pairwise strength of relationship, but not causality
Causality indicates the likely impact of IV on DV
E.g. what is the long-term trend in enrolment in higher education?
It is possible to calculate correlations between a large number of variables
You get a matrix with the same number of rows and columns as the total number of variables
(If you use the partial correlations procedure, you can calculate correlation between two variables, controlling for the effect of a third variable)
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45. Simple regression analysis
Linear regression procedure in SPSS
Analyze
Regression
Linear
Transfer DV and IV(s) to list OK
Note – the DV should be continuous
It is not a condition that the DV has normal distribution
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46. Simple regression analysis
Enrolment = , ,373*Year
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47. Simple regression analysis
Best fit line procedure in SPSS
Analyze
Regression
Curve estimation
Place variables on RHS
Tick linear OK
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48. Simple regression analysis
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49. Summary 2 variables One-Sample T Test Paired Samples T Test
Test if one variable is different from (for example) zero
Paired Samples T Test
Test if two variables in the same sample are different from each other
Independent Samples T Test
Test if (the same) variable(s) in different samples are different from each other
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50. Summary 2 variables Chi-square Test One-Way ANOVA Correlation analysis
Appropriate for discrete data
One-Way ANOVA
The DV must be a ratio variable and continuous
Correlation analysis
Can be run with any kind of data, but is not appropriate for nominal variables with more than 2 categories
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51. Summary 2 variables Simple Regression Analysis Best Fit Line
Same procedure as multiple linear regression analysis, only with fewer variables (just one IV)
Appropriate if DV is continuous
Best Fit Line
Is a simple linear regression analysis under a different name
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52. Multiple regression analysis
Let’s return to the African farmers
What explains whether farmers grow improved or traditional varieties of pigeonpeas?
Available variables:
Farmer characteristics (sex, age, education, household size)
Environmental factors (distance to the nearest main market, district)
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53. Multiple regression analysis
Design issues
There are several regression methods, based on theoretical considerations
In the absence of a strong theoretical reason to choose otherwise, use the standard procedure, i.e., “Enter”
If IVs are correlated, they will generate a variance inflation effect that reduces the statistical significance of results
To check for correlation (“multicollinearity”), we want to run a test along with the regression
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54. Multiple regression analysis
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55. Multiple regression analysis
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56. Multiple regression analysis
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57. Multiple regression analysis
(1 = Female)
Using standardized coefficients, interpretations are based on the standard deviations of the variables.
Each coefficient indicates the number of standard deviations that the predicted response changes for a one standard deviation change in a predictor, all other predictors remaining constant.
A value of Tolerance <0.2 or of VIF >5 indicates presence of multicollinearity.
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58. Multiple regression analysis
Nominal variables are converted to dummies
Sex (binary, 0 = Male, 1 = Female)
Districts
One category is omitted (here: Karatu District)
The others are represented by binary variables (0 = No, 1 = Yes)
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59. Logistic regression The DV is a discrete variable
Binary
Nominal with more than 2 categories
Sensitive to problems like…
Multicollinearity
Small sample size
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60. Logistic regression
Missing observations are farmers with no pigeonpeas at all.
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61. Logistic regression
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62. Logistic regression
The reference category is farmers in Karatu who have planted all of their pp fields in improved varieties
(… and District = Karatu)
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63. Summary ≥ 2 variables Multiple regression analysis Logistic regression
Maps the relationship between one DV and many IVs
DV must be a ratio variable and continuous
IV can be ratio or nominal (dummy)
Logistic regression
Appropriate if the DV is a discrete variable
Sensitive to multicollinearity and small sample size
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