Statistics Basics Alyson Wilson agwilso2@ncsu August 20, 2018

1. Statistics BasicsAlyson Wilsonagwilso2@ncsu.eduAugust 20, 2018

2. Populations and Samples • Population (Big Who) • Group of people we want information from. • Generally, very large. • Impractical or prohibitively expensive to talk to everyone • Sample (Small Who) • Smaller group of people from population. • Group we get information from.

3. What’s a model? • A model is a simple general description of a population • For the univariate models we’re looking at first, there are a couple of ways I often think about how they describe the population: • Data generating mechanism • “Limit”: Draw a larger and larger sample, make your bins for the histogram finer and finer, gets closer to the function . . . .

4. Normal Distribution • Normal(m,s2) • Mean m • s.d.s

5. Poisson Distribution • Unimodal • Center (mean l) • Spread (s.d. ) • Discrete (counts) Model Sample

6. Poisson X counts the number of occurrences across a specified interval. Poisson(l) P[X = x] = x = 0, 1, 2, 3, . . . .; l > 0 E[X] = λ Var[X] = λ

7. What’s a model? When we introduce models that include explanatory variables, we often use those variables to “model” the parameters. For example, linear regression

8. More on Models • The class of models we are considering are called “probability distributions” • Much like data, it is useful to group these models into “discrete” and “continuous” • The models are specified with a small number of parameters

9. More on Models • To identify a particular model, we use its name and the parameter list • Normal(m,s2) • Poisson(l) • Exponential(l) • Some models have one parameter, some two, some three, some a vector of length k

10. More on Models Each model also has two associated functions: the density (continuous) or mass (discrete) function and the cumulative distribution function. The density/mass function is the function that was plotted on the previous slides (e.g., the normal “bell” curve). The cumulative distribution function is calculated from the density/mass function using integration/summation.

11. Inference We have a model for the population The population model has parameters We get a sample from the population We use the sample to calculate estimates of the parameters We associate uncertainty with the estimates of the parameters

12. Be sure to read the next 9 slides on discrete and continuous probability distributions.

13. Discrete Probability Distributions A probability mass function for a discrete random variable X that can take on possible values x1, x2, . . ., is a non-negative function f(x), with f(xi) giving the probability that X takes on the value xi. • f(xi) ≥ 0 • Σ f(xi) = 1

14. Expected Value The mean or expected value of a discrete random variable X is E[X] =

15. Variance The variance of a discrete random variable X is Var[X] = = - E[X]2 The standard deviation of X is

16. Cumulative Distribution Function Cumulative distribution function (cdf): F(x) for the discrete random variable X is defined as the probability that X is less than or equal to x F(x) = P[X ≤ x] =

17. Continuous Probability Distributions Probability density function f(x) ≥ 0 The intuition that we used for discrete random variables that the density function is the probability that X = x breaks down for continuous random variables. Why?

18. Continuous Probability Distributions Instead we can think about either the cumulative distribution function or about the probability that X takes on a value in some interval (a,b)

19. Expected Value The mean or expected value of a continuous random variable X is E[X] = E[g(X)] =

20. Variance The variance of a discrete random variable X is Var[X] = = - E[X]2 = E[X2] – E[X]2 The standard deviation of X is

21. Median The value x0 such that

22. Poisson X counts the number of occurrences across a specified interval. Poisson(l) P[X = x] = x = 0, 1, 2, 3, . . . .; l > 0 E[X] = λ Var[X] = λ

23. Poisson Distribution • Unimodal • Center (mean l) • Spread (s.d. ) • Discrete (counts) Model Sample

24. Estimates • The Poisson distribution has one parameter l. • l is the mean or expected value of the Poisson distribution. We write E[X] = λ, where “X” is our notation for a single draw from the distribution.

25. Estimates • Use R and draw a sample of size 100 from a Poisson distribution with parameter 5. • Draw a barchart(hist()) of the sample. • Calculate the mean and standard deviation of the sample. One simple way to estimate λ is to equate the sample mean to the population mean.

26. Method of Moments kthpopulation moment: E[Xk] kthsample moment: If there are k parameters in the model, we will work with k (population, sample) moment pairs. We will set each pair equal and then solve the equations for the parameters.

27. Method of Moments Method of moments estimators equate sample moments to population moments. 1st population moment: E[X] 1st sample moment: m1 = 2nd population moment: E[X2] 2nd sample moment:

28. Estimating l The Poisson(l) distribution has one parameter. E[X] = l (population moment) m1 = (sample moment) E[X] = m1 → our point estimate of l is the sample mean. We write

29. Estimates • Use R and draw a 10,000 samples of size 100 from a Poisson distribution with parameter 5. • Calculate the sample mean from each sample. • Draw a histogram of the sample means. This is an illustration of sampling variability.

30. sm <- rep(0,10000) for (i in 1:10000) sm[i] <- mean(rpois(100,5)) hist(sm)

31. The Central Limit Theorem • This is an illustration of a very general (and important) result known as the Central Limit Theorem. • If we take a random sample, have independent samples, and we sample less than 10% of the population, then • as our sample size gets large enough (more on the next slide), • the distribution of the sample means is (approximately) normally distributed • with mean equal to the population mean and • standard deviation equal to the (population standard deviation)/(square root of the sample size).

32. How many samples? • It depends on shape of population distribution • Symmetric: 5-15 samples • Skewed: > 25 samples, sometimes many more

33. How does this help us? • We know that if we took lots of samples of size 100 and calculated , the distribution of the sample means would be normally distributed with mean l and standard deviation = population standard deviation/10. • Because of properties of the normal distribution, we know if we go out 2 (actually 1.96) standard deviations to either side of the mean, we will see 95% of the values.

34. We want to say something about the uncertainty in our estimate. • 95% of the time, • Problem: We don’t know the population standard deviation.

35. Confidence Intervals Since we don’t know the population standard deviation, we approximate it with the sample standard deviation. Rewriting the inequalities on each side This is called a 95% confidence interval.

36. Confidence Intervals lb <- rep(0,10000) sm <- rep(0,10000) ssd <- rep(0,10000) ub <- rep(0,10000) inc <- rep(0,10000) for (i in 1:10000) { x <- rpois(100,5) sm[i] <- mean(x) ssd[i] <- sd(x) lb[i] <- sm[i] - 1.96*ssd[i]/10 ub[i] <- sm[i] + 1.96*ssd[i]/10 if ((lb[i] <= 5) & ub[i] >= 5) inc[i] <- 1 } hist(sm) hist(ssd) hist(lb) hist(ub) sum(inc)/10000

37. Confidence Intervals Problem: When we do our analysis, we don’t know whether or not our particular confidence interval contains the population parameter. What we can say is that 95% of samples this size (n) will produce confidence intervals that capture the true proportion parameter.

38. Estimation and Intervals There are lots of ways to calculate estimates and confidence intervals. Different estimates have different properties that we might want: • Easy (possible!) to compute • Unbiased: expected value of estimate equals population parameter • Consistent: As the sample size goes to infinity, the difference between the estimate and the population value goes to zero. We want a confidence interval to have the correct coverage. If we say it is a 95% interval, it should contain the population value 95% of the time.

39. P-value • We have a sample of data. (Assume n = 100 for this example.) • We hypothesize that this data is a sample from a population that can be modeled as a Poisson(l = 5). • We compute the sample mean, which is the statistic we will use to test our hypothesis. • We see that our sample mean is 5.7. • What do we think about our hypothesis now?

40. P-value sm <- rep(0,10000) for (i in 1:10000) sm[i] <- mean(rpois(100,5)) hist(sm)

41. P-value If our hypothesis about the population is true, by the Central Limit Theorem, if we draw lots of samples and calculate sample means, they should look like draws from a Normal(5,sd = sqrt(5)/10 = 0.224). What’s the probability we see a value of 5.7 or bigger?

42. What’s the probability we see a value of 5.7 or bigger? pnorm(5.7,5,0.224) = 0.999111 So the (one-sided) p-value is 1 – 0.999111 = 0.000889 A small p-value means that the test statistic is not very likely to occur if our hypothesis about the population parameter is correct. P-value