SOC 4015 & SOC 5050 - Lecture 05

Lecture slides for Lecture 05 of the Saint Louis University Course Quantitative Analysis: Applied Inferential Statistics. These slides cover the binomial, poisson, and normal distributions and provide an introduction to statistical significance.

Transcript

1. THE DISTRIBUTION OF RANDOM VARIABLES QUANTITATIVE ANALYSIS CHRISTOPHER PRENER, PH.D. FALL 2018 WEEK 05 LECTURE 05
2. AGENDA QUANTITATIVE ANALYSIS / WEEK 05 / LECTURE 05 1. Front Matter 2. Binomial Distribution 3. Poisson Distribution 4. Normal Distribution 5. Statistical Significance 6. Normality Testing 7. Back Matter
3. ⋆ THEME We want to think 
 systematically about the
 likelihood of observing 
 particular outcomes relative to a known set of possible outcomes.
4. 1 FRONT 
 MATTER
5. Lab 04, PS-02, and Lecture Prep 06 are due before the next lecture. There was an update on the final project due today as an issue in your final project GitHub repo (DoeProject if your last name is “Doe”) 1. FRONT MATTER ANNOUNCEMENTS
6. BINOMIAL DISTRIBUTION 2
7. ▸ A sequence of independent trials (n) with constant probability of success at each trial (p) where we are interested in the number of successes (x) 2. BINOMIAL DISTRIBUTION DEFINITION
8. ▸ A sequence of independent trials (n) with constant probability of success at each trial (p) where we are interested in the number of successes (x) 2. BINOMIAL DISTRIBUTION DEFINITION
9. ▸ A sequence of independent trials (n) with constant probability of success at each trial (p) where we are interested in the number of successes (x) ▸ Acronym: • B = binary outcome • I = independence • N = fixed sample size • S = same probability 2. BINOMIAL DISTRIBUTION DEFINITION
10. ▸ A sequence of independent trials (n) with constant probability of success at each trial (p) where we are interested in the number of successes (x) ▸ 
 ▸ 
 2. BINOMIAL DISTRIBUTION EMPIRICAL RULE
19. 2. BINOMIAL DISTRIBUTION R FUNCTIONS stats::dbinom(k, size=n, prob=p) returns probability of observing k successes P(X = k) stats::pbinom(k, size=n, prob=p, lower.tail = TRUE) returns probability of observing k or fewer successes P(X ≤ k) stats::pbinom(k, size=n, prob=p, lower.tail = FALSE) returns probability of observing more than k successes P(X > k)
20. 2. BINOMIAL DISTRIBUTION BINOMIAL WORKFLOW ▸ What is the probability of 10 or fewer successes occurring in an sequence of 100 independent trials with a binary outcome where the probability of success is .25 for each trial? ▸ Is the binomial distribution appropriate?
21. 2. BINOMIAL DISTRIBUTION BINOMIAL WORKFLOW ▸ What is the probability of 10 or fewer successes occurring in an sequence of 100 independent trials with a binary outcome where the probability of success is .25 for each trial? ▸ Is the binomial distribution appropriate?
22. 2. BINOMIAL DISTRIBUTION BINOMIAL WORKFLOW ▸ What is the probability of 10 or fewer successes occurring in an sequence of 100 independent trials with a binary outcome where the probability of success is .25 for each trial? ▸ Is the binomial distribution appropriate? ▸ What is the appropriate R function?
23. 2. BINOMIAL DISTRIBUTION BINOMIAL WORKFLOW ▸ What is the probability of 10 or fewer successes occurring in an sequence of 100 independent trials with a binary outcome where the probability of success is .25 for each trial? ▸ Is the binomial distribution appropriate? ▸ What is the appropriate R function?
24. 2. BINOMIAL DISTRIBUTION R FUNCTIONS stats::dbinom(k, size=n, prob=p) returns probability of observing k successes P(X = k) stats::pbinom(k, size=n, prob=p, lower.tail = TRUE) returns probability of observing k or fewer successes P(X ≤ k) stats::pbinom(k, size=n, prob=p, lower.tail = FALSE) returns probability of observing more than k successes P(X > k)
25. 2. BINOMIAL DISTRIBUTION R FUNCTIONS stats::dbinom(k, size=n, prob=p) returns probability of observing k successes P(X = k) stats::pbinom(k, size=n, prob=p, lower.tail = TRUE) returns probability of observing k or fewer successes P(X ≤ k) stats::pbinom(k, size=n, prob=p, lower.tail = FALSE) returns probability of observing more than k successes P(X > k)
26. 2. BINOMIAL DISTRIBUTION BINOMIAL WORKFLOW ▸ What is the probability of 10 or fewer successes occurring in an sequence of 100 independent trials with a binary outcome where the probability of success is .25 for each trial? ▸ Is the binomial distribution appropriate? ▸ What is the appropriate R function? ▸ What is n? What is k? What is p?
27. 2. BINOMIAL DISTRIBUTION BINOMIAL WORKFLOW ▸ What is the probability of 10 or fewer successes occurring in an sequence of 100 independent trials with a binary outcome where the probability of success is .25 for each trial? ▸ Is the binomial distribution appropriate? ▸ What is the appropriate R function? ▸ What is n? What is k? What is p?
28. 2. BINOMIAL DISTRIBUTION BINOMIAL WORKFLOW ▸ What is the probability of 10 or fewer successes occurring in an sequence of 100 independent trials with a binary outcome where the probability of success is .25 for each trial? ▸ Is the binomial distribution appropriate? ▸ What is the appropriate R function? ▸ What is n? What is k? What is p? > pbinom(10, size=100, prob=.25, lower.tail = TRUE) [1] 0.0001371006
31. p(k or fewer 
 successes) = 
 .0001
33. p(k or fewer 
 successes) = 
 .0001
34. ▸ A trial where there are only two outcomes - “success” and “failure” ▸ Over the long run (law of large numbers), there is a 50% chance of “success” and a 50% chance of “failure” ▸ Jacob Bernoulli was a Swiss mathematician and professor at the University of Basel 2. BINOMIAL DISTRIBUTION BERNOULLI TRIAL 1654-1705
35. ▸ Blaise Pascal was a French mathematician ▸ Described a process (array) of repeated binomial coefficients that were triangular in shape ▸ Known for centuries in China, Persia, and even in Europe 2. BINOMIAL DISTRIBUTION PASCAL’S TRIANGLE 1623-1662
36. ▸ Blaise Pascal was a French mathematician ▸ Described a process (array) of repeated binomial coefficients that were triangular in shape ▸ Known for centuries in China, Persia, and even in Europe 2. BINOMIAL DISTRIBUTION PASCAL’S TRIANGLE YANG HUI’S TRIANGLE (1303)
37. PASCAL’S TRIANGLE 2. BINOMIAL DISTRIBUTION
38. 2. BINOMIAL DISTRIBUTION PASCAL’S TRIANGLE 1 1 1 1 1 2 1 3 3 1
39. 2. BINOMIAL DISTRIBUTION PASCAL’S TRIANGLE 1 1 1 1 1 2 1 3 3 1 0
40. 2. BINOMIAL DISTRIBUTION PASCAL’S TRIANGLE 1 1 1 1 1 2 1 3 3 1
41. 2. BINOMIAL DISTRIBUTION PASCAL’S TRIANGLE 1 1 1 1 1 2 1 3 3 1
42. 2. BINOMIAL DISTRIBUTION GALTON BOX
43. ▸ Sir Francis Galton English statistician ▸ Cousin of Charles Darwin ▸ A eugenicist who coined the phrase “nature versus nurture” ▸ Demonstrated a number of important statistical ideas, including correlation ▸ Invented what is known as the quincunx or “Galton Box” 2. BINOMIAL DISTRIBUTION “BEAN MACHINE” 1822-1911
44. 2. BINOMIAL DISTRIBUTION GALTON BOX
45. 2. BINOMIAL DISTRIBUTION GALTON BOX
46. 2. BINOMIAL DISTRIBUTION GALTON BOX
47. 2. BINOMIAL DISTRIBUTION GALTON BOX
48. 2. BINOMIAL DISTRIBUTION GALTON BOX
49. 2. BINOMIAL DISTRIBUTION THE QUINCUNX http://goo.gl/qKUTsx
50. 2. BINOMIAL DISTRIBUTION THE QUINCUNX
51. 2. BINOMIAL DISTRIBUTION DE MOIVRE–LAPLACE THEOREM ABRAHAM DE MOIRE PIERRE-SIMON LAPLACE
52. 2. BINOMIAL DISTRIBUTION DE MOIVRE–LAPLACE THEOREM ABRAHAM DE MOIRE 1718
57. POISSON DISTRIBUTION 3
58. ▸ French mathematician and physicist ▸ Laplace’s student ▸ Built upon the binomial distribution to describe events that occur with exceptional rarity 3. POISSON DISTRIBUTION SIMÉON POISSON 1781-1840
59. PRUSSIAN HUSSARDS 3. POISSON DISTRIBUTION LADISLAUS BORTKIEWICZ (1898)
60. DEFINITION ▸ Used for events where n is large and p is very small, … ▸ …so small that their product approaches a constant we call lambda (). 3. POISSON DISTRIBUTION
61. ▸ n = a count of independent events ▸ p = probability 3. POISSON DISTRIBUTION Let: DEFINITION ▸ n can occur a (theoretically) infinite number of times. ▸ Its only parameter is (np) - the greek letter lambda.
62. ▸ n = a count of independent events ▸ p = probability 3. POISSON DISTRIBUTION Let: DEFINITION ▸ n can occur a (theoretically) infinite number of times. ▸ Its only parameter is (np) - the greek letter lambda.
71. 3. POISSON DISTRIBUTION R FUNCTIONS stats::dpois(k, lambda=m) returns probability of observing k successes P(X = k) stats::ppois(k, lambda=m, lower.tail = TRUE) returns probability of observing k or fewer successes P(X ≤ k) stats::ppois(k, lambda=m, lower.tail = FALSE) returns probability of observing more than k successes P(X > k)
72. POISSON WORKFLOW ▸ The probability of a car accident at a given intersection is 0.00004. In a typical week, 100,000 cars pass through the intersection. What is the probability of observing 6 car accidents in a single week at this intersection? • Is the poisson distribution appropriate? 3. POISSON DISTRIBUTION
73. POISSON WORKFLOW ▸ The probability of a car accident at a given intersection is 0.00004. In a typical week, 100,000 cars pass through the intersection. What is the probability of observing 6 car accidents in a single week at this intersection? • Is the poisson distribution appropriate? 3. POISSON DISTRIBUTION
74. POISSON WORKFLOW ▸ The probability of a car accident at a given intersection is 0.00004. In a typical week, 100,000 cars pass through the intersection. What is the probability of observing 6 car accidents in a single week at this intersection? • Is the poisson distribution appropriate? • What is the appropriate R function? 3. POISSON DISTRIBUTION
75. POISSON WORKFLOW ▸ The probability of a car accident at a given intersection is 0.00004. In a typical week, 100,000 cars pass through the intersection. What is the probability of observing 6 car accidents in a single week at this intersection? • Is the poisson distribution appropriate? • What is the appropriate R function? 3. POISSON DISTRIBUTION
76. 3. POISSON DISTRIBUTION R FUNCTIONS stats::dpois(k, lambda=m) returns probability of observing k successes P(X = k) stats::ppois(k, lambda=m, lower.tail = TRUE) returns probability of observing k or fewer successes P(X ≤ k) stats::ppois(k, lambda=m, lower.tail = FALSE) returns probability of observing more than k successes P(X > k)
77. 3. POISSON DISTRIBUTION R FUNCTIONS stats::dpois(k, lambda=m) returns probability of observing k successes P(X = k) stats::ppois(k, lambda=m, lower.tail = TRUE) returns probability of observing k or fewer successes P(X ≤ k) stats::ppois(k, lambda=m, lower.tail = FALSE) returns probability of observing more than k successes P(X > k)
78. POISSON WORKFLOW ▸ The probability of a car accident at a given intersection is 0.00004. In a typical week, 100,000 cars pass through the intersection. What is the probability of observing 6 car accidents in a single week at this intersection? • Is the poisson distribution appropriate? • What is the appropriate R function? • What is ? What is k? 3. POISSON DISTRIBUTION
79. POISSON WORKFLOW ▸ The probability of a car accident at a given intersection is 0.00004. In a typical week, 100,000 cars pass through the intersection. What is the probability of observing 6 car accidents in a single week at this intersection? • Is the poisson distribution appropriate? • What is the appropriate R function? • What is ? What is k? 3. POISSON DISTRIBUTION
80. POISSON WORKFLOW ▸ The probability of a car accident at a given intersection is 0.00004. In a typical week, 100,000 cars pass through the intersection. What is the probability of observing 6 car accidents in a single week at this intersection? • Is the poisson distribution appropriate? • What is the appropriate R function? • What is ? What is k? 3. POISSON DISTRIBUTION > dpois(6, lambda=4) [1] 0.1041956
82. p(k successes) = 
 .104
83. NORMAL DISTRIBUTION 4
84. ▸ First suggested by French mathematician Abraham de Moivre as a an outcome of the binomial distribution 4. NORMAL DISTRIBUTION HISTORY 1667-1754
86. ▸ First suggested by French mathematician Abraham de Moivre as a an outcome of the binomial distribution ▸ Carl Friedrich Gauss, a German mathematician who had an immensely influential career, demonstrated its importance in 1809 4. NORMAL DISTRIBUTION HISTORY 1777-1855
87. MEMORIALIZING GAUSS 4. NORMAL DISTRIBUTION
88. ▸ First suggested by French mathematician Abraham de Moivre as a an outcome of the binomial distribution ▸ Carl Friedrich Gauss, a German mathematician who had an immensely influential career, demonstrated its importance in 1809 ▸ Pierre Simon Laplace also made significant contributions to its usefulness beginning in 1810 4. NORMAL DISTRIBUTION HISTORY 1749-1827
89. ▸ As opposed to the binomial and Poisson distributions, the normal is a continuous probability function ▸ Can take on an infinite range of values ( -∞ < x < ∞ ) ▸ Symmetric around , which has same value as median and mode ▸ Spread of distribution determined by ▸ Standard normal has = 0 and 
 = 1 4. NORMAL DISTRIBUTION DEFINITION
102. 68.2% predictive interval
103. 95.4%
104. 99.7%
105. 4. NORMAL DISTRIBUTION Z-SCORES ▸ The value of an observation expressed in standard deviation units. ▸ or stats::pnorm(z, mean=0, sd=1,
 lower.tail=TRUE) returns the cumulative probability under the standard normal distribution P(X ≤ z)
106. ▸ A series of surveys conducted over the past month found that support for a new trade policy was 41.5% with a standard deviation of 2.25. What is the probability that a randomly selected poll shows support for the trade policy to be 46%? • Is the normal distribution appropriate? 4. NORMAL DISTRIBUTION Z-SCORE WORKFLOW
107. ▸ A series of surveys conducted over the past month found that support for a new trade policy was 41.5% with a standard deviation of 2.25. What is the probability that a randomly selected poll shows support for the trade policy to be 46%? • Is the normal distribution appropriate? • What is z? 4. NORMAL DISTRIBUTION Z-SCORE WORKFLOW
108. ▸ A series of surveys conducted over the past month found that support for a new trade policy was 41.5% with a standard deviation of 2.25. What is the probability that a randomly selected poll shows support for the trade policy to be 46%? • Is the normal distribution appropriate? • What is z? 4. NORMAL DISTRIBUTION Z-SCORE WORKFLOW
109. ▸ A series of surveys conducted over the past month found that support for a new trade policy was 41.5% with a standard deviation of 2.25. What is the probability that a randomly selected poll shows support for the trade policy to be 46%? • Is the normal distribution appropriate? • What is z? 4. NORMAL DISTRIBUTION Z-SCORE WORKFLOW > pnorm(2, mean=0, sd=1, lower.tail=TRUE) [1] 0.9772499
112. 97.72%
113. 2.28%
114. STATISTICAL SIGNIFICANCE 5
115. ▸ English mathematician ▸ Student of Sir Francis Galton (and, like Galton, he was a eugenicist and social Darwinist) 5. STATISTICAL SIGNIFICANCE KARL PEARSON 1857-1936
116. ▸ English mathematician ▸ Student of Sir Francis Galton (and, like Galton, he was a eugenicist and social Darwinist) 5. STATISTICAL SIGNIFICANCE KARL PEARSON 1857-1936
117. ▸ English mathematician ▸ Student of Sir Francis Galton (and, like Galton, he was a eugenicist and social Darwinist) ▸ Formalized the concept of the 
 “p-value” around 1914 ▸ Based on Laplace’s earlier use of the idea ▸ Also introduced “moments”, histograms, and a number of other concepts we’ll get to this semester! 5. STATISTICAL SIGNIFICANCE KARL PEARSON 1857-1936
118. ▸ English mathematician and biologist ▸ Like Galton and Pearson, he was a eugenicist and social Darwinist ▸ Introduced the concept of the null hypothesis… ▸ … and popularized the idea of statistical significance including the selection of p = .05. ▸ Responsible for movement away from Bayesian analyses because of his preference for objectivity. 5. STATISTICAL SIGNIFICANCE R.A. FISHER 1890-1962
119. INFORMALLY, A P-VALUE IS THE PROBABILITY UNDER A SPECIFIED STATISTICAL MODEL THAT A STATISTICAL SUMMARY OF THE DATA WOULD BE EQUAL TO OR MORE EXTREME THAN ITS OBSERVED VALUE. American Statistical Association “Statement on p-values”
 (2016)
120. THE PROBABILITY OF GETTING RESULTS AT LEAST AS EXTREME AS THE ONES YOU OBSERVED, GIVEN THAT THE NULL HYPOTHESIS IS CORRECT Christie Aschwanden "Not Even Scientists Can Easily Explain P-values"
 (2015)
121. IMAGINE, HE SAID, THAT YOU HAVE A COIN THAT YOU SUSPECT IS WEIGHTED TOWARD HEADS. (YOUR NULL HYPOTHESIS IS THEN THAT THE COIN IS FAIR.) YOU FLIP IT 100 TIMES AND GET MORE HEADS THAN TAILS. THE P-VALUE WON’T TELL YOU WHETHER THE COIN IS FAIR, BUT IT WILL TELL YOU THE PROBABILITY THAT YOU’D GET AT LEAST AS MANY HEADS AS YOU DID IF THE COIN WAS FAIR. THAT’S IT — NOTHING MORE. Christie Aschwanden "Not Even Scientists Can Easily Explain P-values"
 (2015)
122. 95%
123. 5%
124. 95%
125. 5%
127. z 1.64 2.33 3.09 0.05 0.01 0.001 % of scores < 95% 99% 99.9% % of scores > 5% 1% 0.1%
128. z 1.64 2.33 3.09 p < 0.05 0.01 0.001 % of scores < 95% 99% 99.9% % of scores > 5% 1% 0.1%
129. 95%
130. 2.5%
131. 2.5%
132. 5%
134. z 1.96 2.58 3.29 0.025 0.005 0.0005 % of scores < 97.5% 99.5% 99.95% % of scores > 2.5% 0.5% 0.05%
135. z 1.96 2.58 3.29 0.05 0.01 0.001 % of scores inside 95% 99% 99.9% % of scores outside 5% 1% 0.1%
136. z 1.96 2.58 3.29 p < 0.05 0.01 0.001 % of scores inside 95% 99% 99.9% % of scores outside 5% 1% 0.1%
137. Q: WHY DO SO MANY COLLEGES AND GRAD SCHOOLS TEACH P = 0.05? A: BECAUSE THAT’S STILL WHAT THE SCIENTIFIC COMMUNITY AND JOURNAL EDITORS USE.
 Q: WHY DO SO MANY PEOPLE STILL USE P = 0.05? A: BECAUSE THAT’S WHAT THEY WERE TAUGHT IN COLLEGE OR GRAD SCHOOL. George Cobb, Ph.D. “Statement on p-values”
 (2015)
138. WE TEACH IT BECAUSE IT’S WHAT WE DO; WE DO IT BECAUSE IT’S WHAT WE TEACH. George Cobb, Ph.D. “Statement on p-values”
 (2015)
139. NORMALITY TESTING 6
140. “REAL” DATA ARE NEVER PERFECTLY NORMAL
141. “REAL” DATA ARE NEVER PERFECTLY NORMAL
142. 6. NORMALITY TESTING QUANTIFYING ABNORMAL diagnostic plots descriptive statistics hypothesis tests
143. 6. NORMALITY TESTING QUANTIFYING ABNORMAL diagnostic plots descriptive statistics hypothesis tests
144. ▸ = sample mean ▸ i = lower bound ▸ n = sample size ▸ xi = a given value in the vector 6. NORMALITY TESTING Let: SKEW ¯ x
145. ▸ “Third moment” of the normal distribution ▸ Measure of the asymmetry of a continuous distribution ▸ Normal distributions have sk = 0 ▸ sk > 0 indicates a longer right tail relative to the left tail’s length ▸ sk < 0 indicates a longer left tail relative to the right tail’s length ▸ Values < -2 or > 2 are indicators of a non-normal distribution 6. NORMALITY TESTING SKEW
146. “NORMALLY DISTRIBUTED” HISTOGRAM
147. NON-NORMAL HISTOGRAM
148. NON-NORMAL HISTOGRAM
149. ▸ = sample mean ▸ i = lower bound ▸ n = sample size ▸ xi = a given value in the vector 6. NORMALITY TESTING Let: KURTOSIS ¯ x
150. ▸ “Fourth moment” of the normal distribution ▸ Measure of the “weight” of the tails - how many observations fall in the tails relative to the center ▸ Normal distributions have k = 3 where k is always positive and k ≥ 1. ▸ Normal distributions have k = 0 excess kurtosis (ek = k-3) where ek can be negative. ▸ k > 5 is one rule of thumb for problematic distributions 6. NORMALITY TESTING KURTOSIS
151. 6. NORMALITY TESTING KURTOSIS Mesokurtic - kurtosis = 3 k = 3
152. 6. NORMALITY TESTING KURTOSIS Leptokurtic - kurtosis > 3 k = 4.805
153. 6. NORMALITY TESTING KURTOSIS Platykurtic - kurtosis < 3 k = 1.810
154. 6. NORMALITY TESTING QUANTIFYING ABNORMAL diagnostic plots descriptive statistics hypothesis tests
155. 6. NORMALITY TESTING HISTOGRAM WITH NORMAL DISTRIBUTION
156. QUANTILE-QUANTILE PLOT ▸ Known as the “q-q” plot for short ▸ Plots a given variable against a theoretical distribution (such as the standard normal distribution) as a diagnostic test ▸ Look for how your variable (the points) sits relative to the normal distribution (the 45-degree line). Pay particular attention to the tails. 6. NORMALITY TESTING
157. “NORMALLY DISTRIBUTED”
158. POSITIVE (RIGHT) SKEW
159. NEGATIVE (LEFT) SKEW
160. 6. NORMALITY TESTING QUANTILE-QUANTILE PLOT
161. 6. NORMALITY TESTING QUANTIFYING ABNORMAL diagnostic plots descriptive statistics hypothesis tests
162. 6. NORMALITY TESTING SHAPIRO-FRANCIA TEST 
 Assumptions: 1. Results are valid for 5 ≤ n ≤ 5000
163. 6. NORMALITY TESTING SHAPIRO-FRANCIA TEST Data are not markedly different from the normal distribution. H0 Data are markedly different from the normal distribution. HA
164. 6. NORMALITY TESTING SHAPIRO-FRANCIA TEST Data are not markedly different from the normal distribution. H0 Data are markedly different from the normal distribution. HA If the p value associated with the 
 test statistic is greater than .05…
165. 6. NORMALITY TESTING SHAPIRO-FRANCIA TEST Data are not markedly different from the normal distribution. H0 Data are markedly different from the normal distribution. HA If the p value associated with the 
 test statistic is less than .05…
166. 7 BACK 
 MATTER
167. AGENDA REVIEW 7. BACK MATTER 2. Binomial Distribution 3. Poisson Distribution 4. Normal Distribution 5. Statistical Significance 6. Normality Testing
168. REMINDERS 7. BACK MATTER Lab 04, PS-02, and Lecture Prep 06 are due before the next lecture. There was an update on the final project due today as an issue in your final project GitHub repo (DoeProject if your last name is “Doe”)