HUDM4122 Probability and Statistical Inference. February 2, 2015

Transcription

1 HUDM4122 Probability and Statistical Inference February 2, 2015

2 Special Session on SPSS Thursday, April 23 4pm-6pm As of when I closed the poll, every student except one could make it to this I am happy to meet individually with students who can t make this session

3 And people say pie charts aren t informative From Jeanine DeFalco

4 Homework 1 How did it go? How did you like working with the ASSISTments system? Too few problems? Too many? Just right?

5 What the homework covered Computing the mean, median, mode Symmetric and skewed distributions Variance Standard Deviation

6 Difficulties with rounding Sorry about that I ll try to be clearer next time

7 Difficult Problems 3. You are given n=8 measurements: 3, 2, 5, 6, 4, 4, 3, 4. What is the median? We had answers 4, 4.5, 5 Anyone want to explain any of these answers?

8 Difficult Problems 9. You are given 6 measurements: 5, 4, 4, 6, 8, 6. Calculate the sample variance, s 2 We had answers , 2.3, 2.5, 3.1, 3.3, 11.5 Anyone want to explain any of these answers?

9 Questions? Comments?

10 Beyond these topics, in the last class We looked at how to create and interpret Box Plots And discussed Bimodal Distributions Mean Absolute Deviation Percentiles Z scores

11 Questions? Comments?

12 Today Ch. 3 in Mendenhall, Beaver, & Beaver

13 Today Scatterplots Covariance The Pearson Correlation Coefficient Regression Lines

14 Univariate Data A single variable is collected Height

15 Bivariate Data Two variables are collected (for the same data point) Height Drum-Playing Skill

16 Multivariate Data 3+ variables are collected Name Height Drum-Playing Skill John Lennon Paul McCartney George Harrison Ringo Starr 5 6 8

17 Univariate Data Last Class

18 Bivariate Data Today

19 Scatterplot Shows the relationship between two variables

20 Are more expensive brands of peanut butter better? From InterMath intermath.coe.uga.edu

21 Dependent and Independent Variables Dependent Variable From InterMath intermath.coe.uga.edu Independent Variable

22 The Independent Variable Influences the Dependent Variable (Maybe) Dependent Variable From InterMath intermath.coe.uga.edu Independent Variable

23 (You don t always have to be sure) Dependent Variable From InterMath intermath.coe.uga.edu Independent Variable

24 Data Miners Would Instead Say Predictor and Predicted Variables Predicted Variable From InterMath intermath.coe.uga.edu Predictor Variable

25 I like this terminology better because it s neutral on causation Predicted Variable From InterMath intermath.coe.uga.edu Predictor Variable

26 From InterMath intermath.coe.uga.edu Anyways

27 So which brand of peanut butter should you buy? From InterMath intermath.coe.uga.edu

28 Which brand of peanut butter should a gourmet buy? From InterMath intermath.coe.uga.edu

29 Which brand of peanut butter should a gourmet buy? From InterMath intermath.coe.uga.edu

30 From InterMath intermath.coe.uga.edu How about a frugal person?

31 From InterMath intermath.coe.uga.edu How about a frugal person?

32 Who should buy this peanut butter? From InterMath intermath.coe.uga.edu

33 From InterMath intermath.coe.uga.edu How about this one?

34 From InterMath intermath.coe.uga.edu A lot of variability, right?

35 Questions? Comments?

36 Let s discuss some of the properties of scatterplots

37 What can you say about the relationship between Price and Quality? 300 Snorgles Quality Price

38 What can you say about the relationship between Price and Quality? 250 Frungles 200 Quality Price

39 What can you say about the relationship between Price and Quality? Quality Trandles Price

40 So in other words Spend your hard earned dollars on expensive snorgles But save your money on frungles and trandles

41 Questions? Comments?

42 Quick comment on scatterplots Scatterplots are great

43 Quick comment on scatterplots Scatterplots are great

44 Quick comment on scatterplots But they don t scale to really big data sets If your scatterplot just looks like a giant blob or a grid, try a heat map We won t go into detail on heat maps there s a lot to cover today -- but I wanted to put that in your brains

45 Linear functions All these graphs can be described by linear functions, a.k.a. straight lines

46 Linear functions All these graphs can be described by linear functions, a.k.a. straight lines Snorgles Quality Price

47 Figuring out what the best-fitting line is Is the simplest case of linear regression Linear regression is a sophisticated statistical modeling method Focus of HUDM5122 This is just the simplest application of it

48 Linear Regression: X and Y If the two variables have a linear (straight line) relationship Then we can predict Y s value from X

49 Finding Y from X If you buy a new snorgle that costs $200, what is its quality likely to be? Snorgles Quality Price

50 Finding Y from X If you buy a new snorgle that costs $120, what is its quality likely to be? Snorgles Quality Price

51 There s a better way to do this Snorgles Quality Price

52 We can create a mathematical function Snorgles Quality Price

53 We can create a mathematical function Y= A + BX Slope Y-intercept Snorgles Quality Price

54 In this specific case Quality = A + (B)(Price) Snorgles Quality Price

55 And the exact values are Quality = (Price) Snorgles Quality Price

56 So if you buy a new snorgle for $200, what is its quality likely to be? Quality = (Price) Snorgles Quality Price

57 How was your earlier estimate? Quality = (Price) Snorgles Quality Price

58 So if you buy a new snorgle for $120, what is its quality likely to be? Quality = (Price) Snorgles Quality Price

59 How was your earlier estimate? Quality = (Price) Snorgles Quality Price

60 What we just did is called interpolation We used a formula to find an unknown value of Y For a value of X Where the value of X was between The minimum and maximum known values of the X variable

61 How did I compute A and B? There s at least three answers

62 Answer #1

63 Answer#1: Least Squares Regression Formulas I used the magical Least Squares Regression Formulas = =

64 Answer#2: Least Squares Regression Formulas I computed first order partial derivatives in order to discover the magical Least Squares Regression Formulas Here s a pretty good explanation (if you know Calculus) mjelde-james/ageco317/read/simple-9.doc

65 Answer#3: Minimize the Sum of Squared Residuals I found the values for A and B that made the sum of squared residuals the smallest A residual is how much each predicted value for Y differs from the actual value for Y The formula computes this easily (after derivation), but you can also do this in Excel! If we have time at the end, I ll show you this

66 Questions? Comments?

67 Slope

68 Slope Slope (B) = 1 Slope (B) =

69 Slope Slope (B) = Slope (B) = 0.5 Slope (B) =

70 Slope Goes between positive infinity and negative infinity Can anyone draw Y = 1 1X Y = 1 3X Y = 1 1/3X

71 Questions? Comments?

72 So far we ve looked at very clean data

73 Real data is usually messier

74 Here s some data I downloaded from greatschools.net Every suburban high school in Allegheny County, Pennsylvania Household income of region Percent of students who got top score on state standardized exam

75 100 Percent of Studnets "Advanced" on Exam Average Household Income

76 Formula Y = X 100 Percent of Studnets "Advanced" on Exam Average Household Income

77 What percent advanced for a school with avg income = 30K? Y = X 100 Percent of Studnets "Advanced" on Exam Average Household Income

78 What percent advanced for a school with avg income = 120K? Y = X 100 Percent of Studnets "Advanced" on Exam Average Household Income

79 What we just did is called extrapolation We used a formula to find an unknown value of Y For a value of X Where the value of X was outside The minimum and maximum known values of the X variable

80 Extrapolation can be dangerous As we just saw in the last example

81 Questions? Comments?

82 In the last lecture We looked at the variability of a single variable

83 In the last lecture We looked at the variability of a single variable There can be variability in a relationship too

84 In the last lecture We looked at the variability of a single variable There can be variability in a relationship too

85 In the last lecture We looked at the variability of a single variable There can be variability in a relationship too Stemming from the variability of each of the variables involved

86 In the last lecture We looked at the variability of a single variable There can be variability in a relationship too Stemming from the variability of each of the variables involved

87 From McDonald, 2014,

88 We can measure that variability

89 We can measure that variability Using the correlation coefficient

90 We can measure that variability Using the correlation coefficient Also called Pearson Correlation Pearson Product-Moment Correlation

91 Important Note Even though many people refer to the Pearson correlation as the correlation A correlation is simply any relationship between two or more variables

92 Important Note Even though many people refer to the Pearson correlation as the correlation A correlation is simply any relationship between two or more variables When A s value changes, does B change in the same direction?

93 Beyond Pearson correlation

94 Beyond Pearson correlation (and this class)

95 Beyond Pearson correlation (and this class) Spearman s ρ Kendall s τ Goodman and Kruskal s γ Intraclass correlation

96 Is written r Pearson correlation

97 Pearson correlation Looks at the strength of the linear relationship between two variables

98 Pearson correlation Looks at the strength of the linear relationship between two variables If it s a nonlinear relationship, you want something that can model nonlinear correlations, like Spearman s ρ

99 Pearson correlation Looks at the strength of the linear relationship between two variables Looks at the quality of a linear model of the relationship between two variables

100 Pearson correlation Looks at the strength of the linear relationship between two variables Looks at the quality of a linear model of the relationship between two variables Yes, the Ax + B we just looked at

101 Close relationship Moderate relationship Weak relationship From McDonald, 2014,

102 What is a good correlation? 1.0 perfect 0.0 none -1.0 perfectly negatively correlated In between depends on the field

103 What is a good correlation? 1.0 perfect 0.0 none -1.0 perfectly negatively correlated In between depends on the field In physics correlation of 0.8 is weak! In education correlation of 0.3 is good

104 Scoping Correlations Time on Task and Learning Cigarette Smoking and Lifespan - 0.3

105 From D. Boigelot, Wikipedia Pearson correlation values

106 How do we compute correlation?

107 First we have to compute the covariance Covariance is to Pearson Correlation As Variance is to Standard Deviation

108 First we have to compute the covariance Covariance is to Pearson Correlation As Variance is to Standard Deviation i.e. the same idea, but its values aren t interpretable

109 Written First we have to compute the covariance

110 Written First we have to compute the covariance = ( )( ) ( )

111 First we have to compute the covariance = ( )( ) ( ) In other words Compute means for X and Y Take each deviation for X Take each deviation for Y For each i Multiply the i-th deviation X by the i-th deviation Y Add them all together Divide by n-1

112 What does it mean to multiply deviations together? Note that we re multiplying deviations Not absolute deviations or standard deviations I.e. sometimes the values are positive, sometimes they re negative

113 And also note We re not comparing all values to each other Just the 1 st X to the 1 st Y And the 2 nd X to the 2 nd Y And the 3 rd X to the 3 rd Y And so on

114 So If the 1 st X is really above the mean 1 st Y is really above the mean Then you ll add a big positive number to the covariance ( )( ) ( 1)

115 So If the 1 st X is really below the mean 1 st Y is really below the mean Then you ll add a big positive number to the covariance ( )( ) ( 1)

116 So If the 1 st X is really above the mean 1 st Y is really below the mean Then you ll add a big negative number to the covariance ( )( ) ( 1)

117 So If the 1 st X is really below the mean 1 st Y is really above the mean Then you ll add a big negative number to the covariance ( )( ) ( 1)

118 So If the 1 st X is near the mean 1 st Y is near the mean Then you ll add a number close to zero to the covariance ( )( ) ( 1)

119 Questions? Comments?

120 Let s do an example together Compute means for X and Y Take each deviation for X Take each deviation for Y For each i Multiply the i-th deviation X by the i-th deviation Y Add them all together Divide by n-1 X Y

121 Please do another example in pairs Compute means for X and Y Take each deviation for X Take each deviation for Y For each i Multiply the i-th deviation X by the i-th deviation Y Add them all together Divide by n-1 X Y

122 Questions? Comments?

123 Computing Pearson Correlation =

124 Returning to our example = X Y

125 Please compute the Pearson Correlation in = = ( )( ) ( ) pairs X Y

126 Questions? Comments?

127 Correlations can be vulnerable to outliers

128 Anscombe s Quartet: Same correlation, very different relationships Image from Wikipedia

129 Anscombe s Quartet: Same correlation, very different relationships Noisy Linear Relationship Nonlinear Relationship Outlier Super Outlier! Image from Wikipedia

130 Implication of Anscombe s Quartet Don t just compute a Pearson correlation, see a high value, and declare victory Actually look at a scatterplot

131 So, that s correlation

132 So, that s correlation Useful way to see if two variables are related to each other (linearly)

133 Please remember Correlation is not causation!

134 Questions? Comments?

135 If we have time Demo finding A and B for Y = A + Bx In Excel Using Sum of Squared Residuals Residual = Difference Between Predicted Y and Actual Y

136 Questions? Comments?

137 Upcoming Classes 2/4 Introduction to Probability Ch /9 No class 2/11 Permutations, Combinations, Unions, and Complements Ch. 4.4 HW2 due

138 Homework 2 Due in 9 days In the ASSISTments system

139 Questions? Comments?