URBP 204A QUANTITATIVE METHODS I Statistical Analysis Lecture IV

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URBP 204A QUANTITATIVE METHODS IStatistical
Analysis Lecture IV

• Gregory Newmark
• San Jose State University
• (This lecture is based on Chapters 5,12,13, 15
  of Neil Salkinds
• Statistics for People who (Think They) Hate
  Statistics, 2nd Edition
• which is also the source of many of the offered
  examples. All cartoons are from CAUSEweb.org by
  J.B. Landers.)

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More Statistical Tests

• Factorial Analysis of Variance (ANOVA)
• Tests between means of more than two groups for
  two or more factors (independent variables)
• Correlation Coefficient
• Tests the association between two variables
• One Sample Chi-Square (?2)
• Tests if an observed distribution of frequencies
  for one factor is what one would expect by chance
• Two Factor Chi-Square (?2)
• Tests if an observed distribution of frequencies
  for two factors is what one would expect by chance

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Factorial ANOVA

• Compares observations of a single variable among
  two or more groups which incorporate two or more
  factors.
• Examples
• Reading Skills
• School (Elementary, Middle, High)
• Academic Philosophy (Montessori, Waldorf)
• Environmental Knowledge
• Commute Mode (Car, Bus, Walking)
• Age (Under 40, 40)
• Wealth
• Favorite Team (As, Giants, Dodger, Angels)
• Home Location (Oakland, SF, LA)
• Weight Loss
• Gender (Male, Female)
• Exercise (Biking, Running)

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Factorial ANOVA

• Two Types of Effects
• Main Effects differences within one factor
• Interaction Effects differences across factors
• Example
• Weight Loss
• Gender (Male, Female)
• Exercise (Biking, Running)
• Main Effects
• Does weight loss vary by exercise?
• Does weight loss vary by gender?
• Interaction Effects
• Does weight loss due to exercise vary by gender?

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Factorial ANOVA

• Example
• How is weight loss affected by exercise program
  and gender?
• Steps
• State hypotheses
• Null
• H0 µMale µFemale
• H0 µBiking µRunning
• H0 µMale-Biking µFemale-Biking
  µMale-Running µFemale-Running
• Research
• What would these three be?

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Factorial ANOVA

• Steps (Continued)
• Set significance level
• Level of risk of Type I Error 5
• Level of Significance (p) 0.05
• Select statistical test
• Factorial ANOVA
• Computation of obtained test statistic value
• Insert obtained data into appropriate formula
• (SPSS can expedite this step for us)

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Factorial ANOVA

• Weight Loss Data

Male-Biking Male-Running Female-Biking Female-Running
76 88 65 65
78 76 90 67
76 76 65 67
76 76 90 87
76 56 65 78
74 76 90 56
74 76 90 54
76 98 79 56
76 88 70 54
55 78 90 56
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Factorial ANOVA

• SPSS Outputs

p
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Factorial ANOVA

• SPSS Outputs

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Factorial ANOVA

• SPSS Outputs
• Graph them!

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Factorial ANOVA

• Steps (Continued)
• Computation of obtained test statistic value
• Exercise F 2.444, p 0.127
• Gender F 1.908, p 0.176
• Interaction F 9.683, p 0.004
• Look up the critical F score
• dfnumerator of Factors 1
• dfdenominator of Observations of Groups
• What is the critical F score?
• Comparison of obtained and critical values
• If obtained gt critical reject the null hypothesis
• If obtained lt critical stick with the null
  hypothesis

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Factorial ANOVA

• Steps (Continued)
• Therefore we reject the null hypothesis for the
  interaction effects. This means that while
  choice of exercise alone and gender alone make no
  difference to weight loss, in combination they do
  differentially affect weight loss. Men should
  run and women should bike, according to these
  data.

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Correlation Coefficient

• Tests whether changes in two variables are
  related
• Examples
• Are property values positively related to
  distance from waste dumps?
• Is age correlated with height for minors?
• Are apartment rents negatively related to
  commute time?
• Does someones height relate to income?
• How related are hand size and height?

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Correlation Coefficient

• Are Tastiness and Ease correlated for fruit?
• Is there directionality?

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Correlation Coefficient

• Numeric index that reflects the linear
  relationship between two variables (bivariate
  correlation)
• How does the value of one variable change when
  another variable changes?
• Each case has two data points
• E.g. This study records each persons height and
  weight to see if they are correlated.
• Ranges from -1.0 to 1.0
• Two types of possible correlations
• Change in the same direction positive or direct
  correlation
• Change in opposite directions negative or
  indirect correlation
• Absolute value reflects strength of correlation
• Pearson Product-Moment Correlation
• Both variables need to be ratio or interval

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Correlation Coefficient

• Scatterplot

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Correlation Coefficient

• Coefficient of Determination
• Squaring the correlation coefficient (r2)
• The percentage of variance in one variable that
  is accounted for by the variance in another
  variable
• Example GPA and Time Spent Studying
• rGPA and Study Time 0.70 r2GPA and Study
  Time 0.49
• 49 of the variance in GPA can be explained by
  the variance in studying time
• GPA and studying time share 49 of the variance
  between themselves

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Correlation Coefficient

• Example
• How related are hand size and height?
• Steps
• State hypotheses
• Null H0 ?Hand Size and Height 0
• Research H1 rHand Size and Height ? 0
• Non-directional
• Set significance level
• Level of risk of Type I Error 5
• Level of Significance (p) 0.05

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Correlation Coefficient

• Steps (Continued)
• Select statistical test
• Correlation Coefficient (it is the test
  statistic!)
• Computation of obtained test statistic value
• Insert obtained data into appropriate formula

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Correlation Coefficient

• Plot the data n 30

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Correlation Coefficient

• Steps (Continued)
• Computation of obtained test statistic value
• rHand Size and Height 0.736

Correlations Correlations Correlations Correlations
Height Hand
Height Pearson Correlation 1 .736
Height Sig. (2-tailed) .000
Height N 30 30
Hand Pearson Correlation .736 1
Hand Sig. (2-tailed) .000
Hand N 30 30
. Correlation is significant at the 0.01 level (2-tailed). . Correlation is significant at the 0.01 level (2-tailed). . Correlation is significant at the 0.01 level (2-tailed). . Correlation is significant at the 0.01 level (2-tailed).
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Correlation Coefficient

• Steps (Continued)
• Computation of critical test statistic value
• Value needed to reject null hypothesis
• Look up p 0.05 in critical value table
• Consider degrees of freedom df n 2
• Consider number of tails (is there
  directionality?)
• rcritical ?

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Correlation Coefficient

• What happens to the critical score when the
  number of cases (n) decreases? Why?

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Correlation Coefficient

• Steps (Continued)
• Comparison of obtained and critical values
• If obtained gt critical reject the null hypothesis
• If obtained lt critical stick with the null
  hypothesis
• robtained 0.736 gt rcritical 0.349
• Therefore, we reject the null hypothesis and
  accept the research hypothesis that height and
  handbreadth are correlated.
• Is there a directionality to that correlation?

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Correlation Coefficient

• Significance vs. Meaning
• Rules of Thumb
• r 0.8 to 1.0 Very strong relationship
• r 0.6 to 0.8 Strong relationship
• r 0.4 to 0.6 Moderate relationship
• r 0.2 to 0.4 Weak relationship
• r 0.0 to 0.2 Weak or no relationship

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Correlation Coefficient

• Does correlation express causation?
• Classic Example
• Ice Cream Eaten
• Crimes Committed

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Correlation Coefficient

• Correlation expresses association only

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Chi-Square (?2)

• Non-Parametric Test
• Does not rely on a given distribution
• Useful for small sample sizes
• Enables consideration of data that comes as
  ordinal or nominal frequencies
• Number of children in different grades
• Percentage of people by state receiving social
  security

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One Sample Chi-Square (?2)

• Tests whether an observed distribution of
  frequencies for one factor is likely to have
  occurred by chance
• Examples
• Is this community evenly distributed among
  ethnic groups?
• Are the 31 ice cream flavors at Baskin Robbins
  equally purchased?
• Are commuting mode shares evenly spread out?
• Did people report equal preferences for a school
  voucher policy?

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One Sample Chi-Square (?2)

• Examples
• Did people report equal preferences for a school
  voucher policy?
• Data (90 People split into 3 Categories)
• For 23
• Maybe 17
• Against 50
• Always try to have at least 5 responses per
  category

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One Sample Chi-Square (?2)

• Steps
• State hypotheses
• Null
• H0 ProportionFor ProportionMaybe
  ProportionAgainst
• Research
• H1 ProportionFor ? ProportionMaybe ?
  ProportionAgainst
• Set significance level
• Level of risk of Type I Error 5
• Level of Significance (p) 0.05
• Select statistical test
• Chi-Square (?2)

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One Sample Chi-Square (?2)

• Steps (Continued)
• Computation of obtained test statistic value
• Insert obtained data into appropriate formula
• (SPSS can expedite this step for us)

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One Sample Chi-Square (?2)

• Steps (Continued)
• Computation of obtained test statistic value

Category O E (O-E) (O-E)2 (O-E)2/E
For 23 30 -7 49 1.63
Against 17 30 -13 169 5.63
Maybe 50 30 20 400 13.33
Total 90 90 -- -- 20.59
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One Sample Chi-Square (?2)

• Steps (Continued)
• Computation of obtained test statistic value
• ?2 obtained 20.59
• Computation of critical test statistic value
• Value needed to reject null hypothesis
• Look up p 0.05 in ?2 table
• Consider degrees of freedom df of categories
  - 1
• ?2 critical 5.99

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One Sample Chi-Square (?2)

• Steps (Continued)
• Computation of obtained test statistic value

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One Sample Chi-Square (?2)

• Steps (Continued)
• Comparison of obtained and critical values
• If obtained gt critical reject the null hypothesis
• If obtained lt critical stick with the null
  hypothesis
• ?2 obtained 20.59 gt ?2 critical 5.99
• Therefore, we can reject the null hypothesis and
  we thus conclude that distribution of preferences
  regarding the school voucher is not even.

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Two Factor Chi-Square (?2)

• What if we want to see if gender effects the
  distribution of votes?
• How is this different from Factorial ANOVA?

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Two Factor Chi-Square (?2)

• Steps
• State hypotheses
• Null
• H0 PForMale PMaybeMale PAgainst Male
  PForFemale PMaybeFemale PAgainst Female
• Research
• H1 PForMale ? PMaybeMale ? PAgainst Male ?
  PForFemale ? PMaybeFemale ? PAgainst Female
• Set significance level
• Level of risk of Type I Error 5
• Level of Significance (p) 0.05
• Select statistical test
• Chi-Square (?2)

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Two Factor Chi-Square (?2)

• Steps (Continued)
• Computation of obtained test statistic value
• Insert obtained data into appropriate formula
• Same as for One Factor Chi-Square

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Two Factor Chi-Square (?2)

• How do we find the expected frequencies?
• (Row Total Column Total)/ Total Total
• Expected Value ForMale (2344)/90 11.2

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Two Factor Chi-Square (?2)

• Steps (Continued)
• Computation of obtained test statistic value
• ?2 obtained 7.750

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Two Factor Chi-Square (?2)

• Steps (Continued)
• Computation of critical test statistic value
• Value needed to reject null hypothesis
• Look up p 0.05 in ?2 table
• Consider degrees of freedom
• df ( of rows 1) ( of columns 1)
• ?2 critical ?

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Two Factor Chi-Square (?2)

• Steps (Continued)
• Comparison of obtained and critical values
• If obtained gt critical reject the null hypothesis
• If obtained lt critical stick with the null
  hypothesis
• ?2 obtained 7.750 gt ?2 critical 5.99
• Therefore, we can reject the null hypothesis and
  we thus conclude that gender affects the
  distribution of preferences regarding the school
  vouchers.

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Tutorial Time