Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Introduction to Basic Statistical Methods Part 1: “Statistics in a Nutshell” UWHC Scholarly Forum March 19, 2014 Ismor Fischer, Ph.D. UW Dept of Statistics [email protected] UWHC Scholarly Forum March 19, 2014 Ismor Fischer, Ph.D. UW Dept of Statistics [email protected] All slides posted at http://www.stat.wisc.edu/~ifischer/Intro_Stat/UWHC • Right-cick on image for full .pdf article • Links in article to access datasets “Statistical Inference” POPULATION Women in the U.S. who have given birth “Statistical Inference” POPULATION Study Question: Has mean (i.e., average) of X = “Age at First Birth” of women in the U.S. changed since 2010 (25.4 yrs old)? Present Day: Assume X = “Age at First Birth” follows a normal distribution (i.e., “bell curve”) in the population. Population Distribution X But what does that mean (at least in principle)? “Statistical Inference” POPULATION Study Question: Has mean (i.e., average) of X = “Age at First Birth” of women in the U.S. changed since 2010 (25.4 yrs old)? Present Day: Assume X = “Age at First Birth” follows a normal distribution (i.e., “bell curve”) in the population. Population Distribution x4 x5 x2 x1 X x3 Individual ages from the population tend to collect around a single center with a certain amount of spread, but occasional “outliers” are present in left and right symmetric tails. More precisely… ad infinitum… ~ The Normal Distribution ~ “population standard deviation” f ( x) symmetric about its mean unimodal (i.e., one peak), with left and right “tails” models many (but not all) naturally-occurring systems useful mathematical properties… “population mean” Example: X = Body Temp (°F) low variability small 98.6 ~ The Normal Distribution ~ “population standard deviation” f ( x) symmetric about its mean unimodal (i.e., one peak), with left and right “tails” models many (but not all) naturally-occurring systems useful mathematical properties… “population mean” Example: X = Body IQ score Temp (°F) low high variability small large 98.6 100 ~ The Normal Distribution ~ “population standard deviation” 95% 2.5% ≈2σ 2.5% ≈2σ f ( x) symmetric about its mean unimodal (i.e., one peak), with left and right “tails” models many (but not all) naturally-occurring systems useful mathematical properties… “population mean” Approximately 95% of the population values are contained between – 2σ and + 2 σ. 95% is called the confidence level. 5% is called the significance level. POPULATION Study Question: Has “Mean (i.e., average) Age at First Birth” of women in the U.S. changed since 2010 (25.4 yrs old)? “Statistical Inference” via… “Hypothesis Testing” Present Day: Assume “Mean Age at First Birth” follows a normal distribution (i.e., “bell curve”) in the population. Population Distribution X cannot be found with 100% certainty, but can be estimated with high confidence (e.g., 95%). H0: pop mean age = 25.4 (i.e., no change since 2010) “Null Hypothesis” POPULATION Study Question: Has “Mean (i.e., average) Age at First Birth” of women in the U.S. changed since 2010 (25.4 yrs old)? “Statistical Inference” via… “Hypothesis Testing” Present Day: Assume “Mean Age at First Birth” follows a normal distribution (i.e., “bell curve”) in the population. Population Distribution X T-test x2 “Null Hypothesis” x4 x1 x3 x5 … etc… x400 H0: pop mean age = 25.4 (i.e., no change since 2010) FORMULA sample mean age x 25.6 x1 x2 x n xn Do the data tend to support or refute the null hypothesis? Is the difference STATISTICALLY SIGNIFICANT, at the 5% level? ~ The Normal Distribution ~ Population Distribution (of ages) “Sampling Distribution” (of mean ages) X n Actually, this is a special case of… via mathematical proof… x2 x1 x3 x4 x5 X … etc… ~ The Normal Distribution ~ Population Distribution (of ages) “Sampling Distribution” (of mean ages) X n Actually, this is a special case of… … as n gets larger CENTRAL LIMIT THEOREM x2 x1 x3 x4 x5 X … etc… ~ The Normal Distribution ~ Population Distribution (of ages) “Sampling Distribution” (of mean ages) X n The sample mean values have much less variability about than the population values! X ~ The Normal Distribution ~ Population Distribution (of ages) “Sampling Distribution” (of mean ages) 95% 2.5% ≈2σ 2.5% n ≈2σ Approximately 95% of the population values are contained between – 2 σ and + 2 σ. Approximately 95% of the sample mean values are contained between 2 n and 2 n X Approximately 95% of the sample mean values are contained between and 2 n 2 n Sample 1 Sample 2 Sample 3 Sample 4 Sample 5 etc… n is called the 95% margin of error X x1 2 x2 x3 x4 x5 Approximately 95% of the sample mean values are contained between and 2 n 2 n Sample 1 Sample 2 Sample 3 Sample 4 Sample 5 n is called the 95% margin of error X x1 2 x2 x3 x4 x5 Approximately Approximately95% 95%of ofthe theintervals sample mean contained x 2values n arefrom x 2between n to and 2 n 2 n contain , and approx 5% do not. Sample 1 Sample 2 Sample 3 Sample 4 Sample 5 n is called the 95% margin of error X x1 2 x2 x3 x4 x5 ~ The Normal Distribution ~ Population Distribution (of ages) “Sampling Distribution” (of mean ages) 95% 2.5% ≈2σ 2.5% n ≈2σ Approximately 95% of the population values are contained between – 2 σ and + 2 σ. Approximately 95% of the sample mean values are contained between 2 n and 2 n Approximately 95% of the intervals x 2 n from x 2 n to contain , and approx 5% do not. X POPULATION Study Question: Has “Mean (i.e., average) Age at First Birth” of women in the U.S. changed since 2010 (25.4 yrs old)? “Statistical Inference” via… “Hypothesis Testing” Present Day: Assume “Mean Age at First Birth” follows a normal distribution (i.e., “bell curve”) in the population. “Null Hypothesis” H0: pop mean age = 25.4 (i.e., no change since 2010) FORMULA SAMPLE n = 400 ages sample mean x x4 x1 x2 x1 x2 n xn = 25.6 x3 x5 … etc… Approximately x95% PROBLEM! 400 of the intervals x 2 n from x 2 n σ is unknown the vast to majority of the time! contain , and approx 5% do not. 95% margin of error 2 n POPULATION Study Question: Has “Mean (i.e., average) Age at First Birth” of women in the U.S. changed since 2010 (25.4 yrs old)? “Statistical Inference” via… “Hypothesis Testing” Present Day: Assume “Mean Age at First Birth” follows a normal distribution (i.e., “bell curve”) in the population. “Null Hypothesis” H0: pop mean age = 25.4 (i.e., no change since 2010) FORMULA SAMPLE n = 400 ages sample mean x x4 x1 x2 x3 x5 … etc… x400 x1 x2 n xn = 25.6 sample variance = modified average of the squared deviations from the mean sample standard deviation 95% margin of error 2 n “Statistical Inference” via… “Hypothesis Testing” POPULATION Study Question: Has “Mean (i.e., average) Age at First Birth” of women in the U.S. changed since 2010 (25.4 yrs old)? Present Day: Assume “Mean Age at First Birth” follows a normal distribution (i.e., “bell curve”) in the population. “Null Hypothesis” H0: pop mean age = 25.4 (i.e., no change since 2010) FORMULA SAMPLE n = 400 ages sample mean x x4 x1 x2 x1 x2 n xn = 25.6 sample variance x3 x5 … etc… x400 ( x1 x ) 2 ( x2 x )2 s n 1 ( xn x ) 2 2 sample standard deviation s s = 1.6 2 95% margin of error 2 n 2 s = 0.16 n Approximately 95% of the intervals x 2 n from x 2 n to contain , and approx 5% do not. x = 25.6 95% margin of error 2 25.44 s = 0.16 n 2 x = 25.6 s = 0.16 n 25.76 BASED ON OUR SAMPLE DATA, the true value of μ today is between 25.44 and 25.76 years, with 95% “confidence” (…akin to “probability”). Two main ways to conduct a formal hypothesis test: 95% CONFIDENCE INTERVAL FOR µ = 25.4 25.44 x = 25.6 25.76 BASED ON OUR SAMPLE DATA, the true value of μ today is between 25.44 and 25.76 years, with 95% “confidence” (…akin to “probability”). IF H0 is true, then we would expect a random sample mean x that is at least 0.2 years away from = 25.4 (as ours was), to occur with probability 1.28%. “P-VALUE” of our sample Very informally, the p-value of a sample is the probability (hence a number between 0 and 1) that it “agrees” with the null hypothesis. Hence a very small p-value indicates strong evidence against the null hypothesis. The smaller the p-value, the stronger the evidence, and the more “statistically significant” the finding (e.g., p < .0001). 25.4 25.6 Two main ways to conduct a formal 95% CONFIDENCE INTERVAL FOR µ hypothesis test: CONCLUSIONS: FORMAL The 95% confidence interval corresponding to our sample mean does not =value” 25.4 of25.44 x = 25.6 contain the “null the population mean, μ = 25.4 years. 25.76 The p-value ourSAMPLE sample,DATA, .0128,the is less predetermined α = .05 BASED ON of OUR truethan valuethe of μ today is between significance 25.44 andlevel. 25.76 years, with 95% “confidence” (…akin to “probability”). Based on our sample data, we may (moderately) reject the null hypothesis is true, expect a alternative random sample mean xH that is at least H0: IFμ H=0 25.4 in then favorwe of would the two-sided hypothesis A: μ ≠ 25.4, 0.2 αyears from =level. 25.4 (as ours was), to occur with probability 1.28%. at the = .05away significance “P-VALUE” of our sample INTERPRETATION: According to the results of this study, there exists a statistically significant difference between the mean ages at first birth in Very informally, theatp-value a sample islevel. the probability 2010 (25.4 years old) and today, the 5%of significance Moreover,(hence the a between 0 and suggest 1) that itthat “agrees” with the null hypothesis. evidence from number the sample data would the population mean age Henceolder a very small p-value indicates evidence against the today is significantly than in 2010, rather thanstrong significantly younger. null hypothesis. The smaller the p-value, the stronger the evidence, and the more “statistically significant” the finding (e.g., p < .0001). However, one problem remains… 25.4 25.6 Normal Distribution Population Distribution (of ages) “Sampling Distribution” (mean ages) 95% 2.5% ≈2σ Normal Distribution 2.5% ≈2σ s n n Approximately 95% of the population values are contained between – 2 σ and + 2 σ. Approximately 95% of the sample mean values are contained between 2 n and 2 n Approximately 95% of the intervals x 2 n from x 2 n to contain , and approx 5% do not. X Normal Distribution Population Distribution (of ages) “Sampling Distribution” (mean ages) 95% 2.5% ≈2σ Normal Distribution 2.5% ≈2σ Approximately 95% of the population values are contained between – 2 s and + 2 s. s n n …IF n is large, e.g., 30 Alas, this introduces “sampling variability.” Approximately 95% of the sample mean values are contained between and 2s n 2s n Approximately 95% of the intervals x 2 s n from x 2 s n to contain , and approx 5% do not. X Edited R code: y = rnorm(400, 0, 1) z = (y - mean(y)) / sd(y) x = 25.6 + 1.6*z Generates a normally-distributed random sample of 400 age values. sort(round(x, 1)) [1] 19.6 20.2 20.4 20.5 21.2 22.3 22.3 22.4 22.4 22.4 22.6 22.7 22.7 22.7 22.8 [16] 23.0 23.0 23.1 23.1 23.2 23.2 23.2 23.2 23.2 23.3 23.4 23.4 23.4 23.5 23.5 etc... [391] 28.7 28.7 28.9 29.2 29.3 29.4 29.6 29.7 29.9 30.2 c(mean(x), sd(x)) [1] 25.6 Calculates sample mean and standard deviation. 1.6 t.test(x, mu = 25.4) One Sample t-test data: x t = 2.5, df = 399, p-value = 0.01282 alternative hypothesis: true mean is not equal to 25.4 95 percent confidence interval: 25.44273 25.75727 sample estimates: mean of x 25.6 Normal Distribution Population Distribution (of ages) “Sampling Distribution” (mean ages) 95% 2.5% ≈2σ Normal Distribution 2.5% ≈2σ Approximately 95% of the population values are contained between – 2 s and + 2 s. s n n …IF n is large, e.g., 30 But if n is small… Approximately 95% of the sample mean values are contained between and 2s n 2s n Approximately 95% of the intervals x 2 s n from x 2 s n to contain , and approx 5% do not. X If n is small, T-score > 2. … the “T-score" increases (from ≈ 2 to a max of 12.706 for a 95% confidence level) as n decreases larger margin of error less power to reject, even if a genuine statistically significant difference exists! If n is large, T-score ≈ 2. POPULATION Study Question: Has “Mean (i.e., average) Age at First Birth” of women in the U.S. changed since 2010 (25.4 yrs old)? “Statistical Inference” via… “Hypothesis Testing” Present Day: Assume “Mean Age at First Birth” follows a normal distribution (i.e., “bell curve”) in the population. T-test x2 “Null Hypothesis” x4 x1 x3 x5 … etc… x400 H0: pop mean age = 25.4 (i.e., no change since 2010) FORMULA sample mean age x 25.6 x1 x2 x n xn Do the data tend to support or refute the null hypothesis? Is the difference STATISTICALLY SIGNIFICANT, at the 5% level? POPULATION Study Question: Has “Mean (i.e., average) Age at First Birth” of women in the U.S. changed since 2010 (25.4 yrs old)? “Statistical Inference” via… “Hypothesis Testing” Present Day: Assume “Mean Age at First Birth” follows a normal distribution (i.e., “bell curve”) in the population. T-test H0: pop mean age = 25.4 (i.e., no change since 2010) “Null Hypothesis” Check? The reasonableness of the normality assumption is empirically verifiable, and in fact formally testable from the sample data. If violated (e.g., skewed) or inconclusive (e.g., small sample size), then “distribution-free” nonparametric tests can be used instead of the T-test. Examples: Sign Test, Wilcoxon Signed Rank Test (= Mann-Whitney Test) POPULATION Study Question: Has “Mean (i.e., average) Age at First Birth” of women in the U.S. changed since 2010 (25.4 yrs old)? “Statistical Inference” via… “Hypothesis Testing” Present Day: Assume “Mean Age at First Birth” follows a normal distribution (i.e., “bell curve”) in the population. T-test x2 “Null Hypothesis” x4 x1 H0: pop mean age = 25.4 (i.e., no change since 2010) x3 x5 … etc… x400 Sample size n partially depends on the power of the test, i.e., the desired probability of correctly rejecting a false null hypothesis (80% or more). Introduction to Basic Statistical Methods Part 1: Statistics in a Nutshell Part 2: Overview of Biostatistics: “Which Test Do I Use??” Sincere thanks to… UWHC Scholarly Forum March 19, 2014 • Judith Payne Ismor Fischer, Ph.D. UW Dept of Statistics [email protected] • Samantha Goodrich • Heidi Miller • Troy Lawrence • YOU!