1. Introduction to Basic Statistical Methods Part 1: Statistics in a Nutshell UWHC Scholarly Forum May 21, 2014 Ismor Fischer, Ph.D. UW Dept of Statistics ifischer@wisc.edu Part 2: Overview of Biostatistics: “Which Test Do I Use??” All slides posted at http://www.stat.wisc.edu/~ifischer/Intro_Stat/UWHC http://www.stat.wisc.edu/~ifischer/Intro_Stat/UWHC

2. Right-cick on image for full.pdf article Links in article to access datasets

3. 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 “Statistical Inference” POPULATION

4. 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 POPULATION “Statistical Inference”

5. ~ The Normal Distribution ~  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” “population standard deviation” 

6. ~ The Normal Distribution ~ “population standard deviation”  symmetric about its mean  unimodal (i.e., one peak), with left and right “tails”  models many (but not all) naturally-occurring systems Approximately 95% of the population values are contained between  – 2 σ and  + 2 σ. 95% is called the confidence level. 5% is called the significance level. 95% 2.5% ≈ 2 σ “population mean”   useful mathematical properties…

7. 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)? x1x1 x4x4 x3x3 x2x2 x5x5 x 400 … etc… “Statistical Inference” Present Day: Assume “Mean Age at First Birth” follows a normal distribution (i.e., “bell curve”) in the population. Population Distribution X “Null Hypothesis” H 0 : pop mean age  = 25.4 (i.e., no change since 2010) via… “Hypothesis Testing”  cannot be found with 100% certainty, but can be estimated with high confidence (e.g., 95%) from sample data. Sample size n partially depends on the power of the test, i.e., the desired probability of correctly rejecting a false null hypothesis (  80%).

8. POPULATION “Null Hypothesis” via… “Hypothesis Testing” Study Question: Has “Mean (i.e., average) Age at First Birth” of women in the U.S. changed since 2010 (25.4 yrs old)? x1x1 x4x4 x3x3 x2x2 x5x5 x 400 … etc… H 0 : pop mean age  = 25.4 (i.e., no change since 2010) “Statistical Inference” Present Day: Assume “Mean Age at First Birth” follows a normal distribution (i.e., “bell curve”) in the population. Population Distribution X sample mean age sample variance

9. sample standard deviation POPULATION “Null Hypothesis” via… “Hypothesis Testing” Study Question: Has “Mean (i.e., average) Age at First Birth” of women in the U.S. changed since 2010 (25.4 yrs old)? x1x1 x4x4 x3x3 x2x2 x5x5 x 400 … etc… H 0 : pop mean age  = 25.4 (i.e., no change since 2010) “Statistical Inference” Present Day: Assume “Mean Age at First Birth” follows a normal distribution (i.e., “bell curve”) in the population. Population Distribution X sample mean age  s = 1.6

10. POPULATION “Null Hypothesis” via… “Hypothesis Testing” Study Question: Has “Mean (i.e., average) Age at First Birth” of women in the U.S. changed since 2010 (25.4 yrs old)? x1x1 x4x4 x3x3 x2x2 x5x5 x 400 … etc… H 0 : pop mean age  = 25.4 (i.e., no change since 2010) “Statistical Inference” Present Day: Assume “Mean Age at First Birth” follows a normal distribution (i.e., “bell curve”) in the population. Population Distribution X sample mean age  s Is the difference STATISTICALLY SIGNIFICANT, at the 5% level? Do the data tend to support or refute the null hypothesis? The population distribution of X follows a bell curve, with standard deviation . = 1.6

11. POPULATION “Null Hypothesis” via… “Hypothesis Testing” Study Question: Has “Mean (i.e., average) Age at First Birth” of women in the U.S. changed since 2010 (25.4 yrs old)? x1x1 x4x4 x3x3 x2x2 x5x5 x 400 … etc… H 0 : pop mean age  = 25.4 (i.e., no change since 2010) “Statistical Inference” Present Day: Assume “Mean Age at First Birth” follows a normal distribution (i.e., “bell curve”) in the population. Population Distribution X sample mean age  s Is the difference STATISTICALLY SIGNIFICANT, at the 5% level? Do the data tend to support or refute the null hypothesis? The “sampling distribution” of also follows a bell curve, with standard deviation  / = 1.6

12. POPULATION “Null Hypothesis” via… “Hypothesis Testing” Study Question: Has “Mean (i.e., average) Age at First Birth” of women in the U.S. changed since 2010 (25.4 yrs old)? x1x1 x4x4 x3x3 x2x2 x5x5 x 400 … etc… H 0 : pop mean age  = 25.4 (i.e., no change since 2010) “Statistical Inference” Present Day: Assume “Mean Age at First Birth” follows a normal distribution (i.e., “bell curve”) in the population. Population Distribution X sample mean age  s Is the difference STATISTICALLY SIGNIFICANT, at the 5% level? Do the data tend to support or refute the null hypothesis? But estimating  by s introduces an additional layer of “sampling variability.” = 1.6

13. POPULATION “Null Hypothesis” via… “Hypothesis Testing” Study Question: Has “Mean (i.e., average) Age at First Birth” of women in the U.S. changed since 2010 (25.4 yrs old)? x1x1 x4x4 x3x3 x2x2 x5x5 x 400 … etc… H 0 : pop mean age  = 25.4 (i.e., no change since 2010) “Statistical Inference” Present Day: Assume “Mean Age at First Birth” follows a normal distribution (i.e., “bell curve”) in the population. Population Distribution X sample mean age  s Is the difference STATISTICALLY SIGNIFICANT, at the 5% level? Do the data tend to support or refute the null hypothesis? In order to take this into account, a cousin to the normal distribution called the “T-distribution” is used instead (Gossett, 1908). = 1.6

14. t1t1 “standard” bell curve:  = 0,  = 1 t df Student’s T-Distribution William S. Gossett (1876 - 1937) … is actually a family of distributions, indexed by the degrees of freedom df = n – 1, labeled t df. As n gets large, t df converges to the standard normal distribution. But the heavier tails mean a wider interval is needed to capture 95%, especially if n is small.

15. T-test POPULATION “Null Hypothesis” via… “Hypothesis Testing” Study Question: Has “Mean (i.e., average) Age at First Birth” of women in the U.S. changed since 2010 (25.4 yrs old)? x1x1 x4x4 x3x3 x2x2 x5x5 x 400 … etc… H 0 : pop mean age  = 25.4 (i.e., no change since 2010) “Statistical Inference” Present Day: Assume “Mean Age at First Birth” follows a normal distribution (i.e., “bell curve”) in the population. Population Distribution X sample mean age  s Is the difference STATISTICALLY SIGNIFICANT, at the 5% level? Do the data tend to support or refute the null hypothesis? In order to take this into account, a cousin to the normal distribution called the “T-distribution” is used instead (Gossett, 1908). = 1.6

16. POPULATION “Null Hypothesis” via… “Hypothesis Testing” Study Question: Has “Mean (i.e., average) Age at First Birth” of women in the U.S. changed since 2010 (25.4 yrs old)? x1x1 x4x4 x3x3 x2x2 x5x5 x 400 … etc… H 0 : pop mean age  = 25.4 (i.e., no change since 2010) “Statistical Inference” Present Day: Assume “Mean Age at First Birth” follows a normal distribution (i.e., “bell curve”) in the population. Population Distribution X sample mean age  s Is the difference STATISTICALLY SIGNIFICANT, at the 5% level? Do the data tend to support or refute the null hypothesis? T-test = 1.6

17. 95% CONFIDENCE INTERVAL FOR µ “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). Two main ways to conduct a formal hypothesis test: 25.7625.44 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”).

18. 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). 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. 25.7625.44 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”). 95% CONFIDENCE INTERVAL FOR µ IF H 0 is true, then we would expect a random sample mean that is at least 0.2 years away from  = 25.4 (as ours was), to occur with probability 1.28%. Two main ways to conduct a formal hypothesis test: “P-VALUE” of our sample FORMAL CONCLUSIONS:  The 95% confidence interval corresponding to our sample mean does not contain the “null value” of the population mean, μ = 25.4 years.  The p-value of our sample,.0128, is less than the predetermined α =.05 significance level. Based on our sample data, we may (moderately) reject the null hypothesis H 0 : μ = 25.4 in favor of the two-sided alternative hypothesis H A : μ ≠ 25.4, at the α =.05 significance level. INTERPRETATION: According to the results of this study, there exists a statistically significant difference between the mean ages at first birth in 2010 (25.4 years old) and today, at the 5% significance level. Moreover, the evidence from the sample data would suggest that the population mean age today is significantly older than in 2010, rather than significantly younger. FORMAL CONCLUSIONS:  The 95% confidence interval corresponding to our sample mean does not contain the “null value” of the population mean, μ = 25.4 years.  The p-value of our sample,.0128, is less than the predetermined α =.05 significance level. Based on our sample data, we may (moderately) reject the null hypothesis H 0 : μ = 25.4 in favor of the two-sided alternative hypothesis H A : μ ≠ 25.4, at the α =.05 significance level. INTERPRETATION: According to the results of this study, there exists a statistically significant difference between the mean ages at first birth in 2010 (25.4 years old) and today, at the 5% significance level. Moreover, the evidence from the sample data would suggest that the population mean age today is significantly older than in 2010, rather than significantly younger.

20. Edited R code: y = rnorm(400, 0, 1) z = (y - mean(y)) / sd(y) x = 25.6 + 1.6*z 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 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 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 Generates a normally-distributed random sample of 400 age values. Calculates sample mean and standard deviation.

21. 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)? x1x1 x4x4 x3x3 x2x2 x5x5 x 400 … etc… “Statistical Inference” Present Day: Assume “Mean Age at First Birth” follows a normal distribution (i.e., “bell curve”) in the population. Population Distribution X H 0 : pop mean age  = 25.4 (i.e., no change since 2010) via… “Hypothesis Testing” Assume 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 should be used instead of the T-test… Examples: Sign Test, Wilcoxon Signed Rank Test (= Mann-Whitney U Test)