Research method ch07 statistical methods 1
- 1. 1 Research Methods in Health Chapter 7. Statistical Methods 1 Young Moon Chae, Ph.D. Graduate School of Public Health Yonsei University, Korea [email protected]
- 2. 2 Topics • What is Biostatistics? • Biostatistics in Public Health Research • Descriptive statistics • Inference statistics • Power of test • T-test
- 3. 3 Concepts Biostatistics • Biostatistics is the development and application of statistics to research in health-related fields. Statistics • Common perceptions of statistics: numbers, tables, figures, polls, rates, etc. • These are “descriptions of the world” • Analysis of data
- 4. 4 Biostatistics in Public Health Research Methodological research: • new statistical techniques • high speed of computing • geographical patterns of disease • clinical trials • longitudinal analysis • data analysis in epidemiology studies
- 5. 5 Errors in Statistical methods • Research design -Improper control group in case-control design -Selection bias (sample does not represent study population) -Too small sample size • Statistical methods -Parametric statistics for small sample -T-test for the related sample -T-test or ANOVA for the samples that do not meet assumptions (normality, equal variances, independence) -T-test for multiple comparison -Regression for nominal dependent variable -Regression with multi-colinearity -Chi-square test with cell size less than 5
- 6. 6 6 Overview of statistical methods
- 7. 7 Descriptive vs. Inferential Statistics • The mean and standard deviation can be used in 2 ways. -One way is to describe the distribution of data -The other way is to infer something about a population (is the population mean 25? 20?). A statistical test! • Because the sampling distribution of the mean is normally distributed (Central Limit Theorem), we can use the normal to show how close the parameter is likely to be to the sample mean and to make decisions about treatments.
- 8. 8 Descriptive Statistics • Descriptive Statistics -Mean, median, mode -Variance, standard deviation, range, Interquartile range, quartile range, -Skewness, Kurtosis • Frequency tables, Bar charts and pie charts, Histograms, Stem-and-Leaf display
- 9. 9 Variables have distributions • A variable is something that changes or has different values (e.g., anger). • A distribution is a collection of measures, usually across people. • Distributions of numbers can be summarized with numbers (called statistics or parameters).
- 10. 10 Central Tendency Central Tendency refers to the Middle of the Distribution
- 11. 11 Middle of the Distribution Common Statistics • Mode -Most common score • Median -Top from bottom 50 percent • Mean -Arithmetic mean or average
- 12. 12 Mode • The most frequently occurring score. Can have bimodal and multimodal distributions. Modal public health student is female.
- 13. 13 Median • Score that separates top 50% from bottom 50% • Even number of scores, median is half way between two middle scores. -1 2 3 4 | 5 6 7 8 – Median is 4.5 • Odd number of scores, median is the middle number -1 2 3 4 5 6 7 – Median is 4
- 14. 14 Mean • Sum of scores divided by the number of people. Population mean is (mu) and sample mean is • We calculate the sample mean by: • We calculate the population mean by: m X N X X å= N Xå=m
- 15. 15 Comparison of statistics • Mode -Good for nominal variables -Good if you need to know most frequent observation -Quick and easy • Median -Good for “bad” (skewed) distributions -Often used with distributions of money • Mean -Used for inference as well as description; best estimator of the parameter -Based on all data in the distribution -Generally preferred except for “bad” distribution. -Most commonly used statistic for central tendency.
- 16. 16 Effects of Distribution Shape
- 17. 17 Distribution Shapes • Normal • Center • Spread • Shoulders • Skew
- 18. 18 Normal
- 20. 20 Variability (spread) Central tendency and Variability
- 23. 23 Inferential Statistics • Estimation: This includes point and interval estimation of certain characteristics in the population(s). • Testing Hypothesis about population parameter(s) based on the information contained in the sample(s).
- 24. 24 Estimation of Parameters • Point Estimation • Interval Estimation (Confidence Intervals) • Bound on the error of estimation • The width of a confidence interval is directly related to the bound on the error.
- 25. 25 Sampling Distribution • Sampling distribution is a distribution of a statistic (not raw data) over all possible samples. Same as distribution over infinite number of trials. • Notion of trials, experiments, replications • Coin toss example (5 flips, # heads) • Repeated estimation of the mean
- 26. 26 Mean of Sampling Distribution • Statisticians have worked out properties of sampling distributions • Middle and spread of sampling distribution are known. • If mean of sampling distribution equals parameter, statistic is unbiased. (otherwise, it’s biased.) The sample mean is unbiased. • Best estimate of is . X Xm
- 27. 27 SD of Sampling Distribution • The standard deviation of the sampling distribution is the standard error. For the mean, it indicates the average distance of the statistic from the parameter. 80787674727068666462605856545250 Heignt in Inches Raw Data Means (N=50) Standard Error Standard error of the mean.
- 28. 28 Factors influencing the Bound on the error of estimation • Narrow confidence intervals are preferred • As the sample size increases, the bound on the error of estimation decreases. • As the confidence level increases the bound on the error of estimation increases. • You need to plan a sample size to achieve the desired level of error and confidence.
- 29. 29 Decision Making Under Uncertainty • You have to make decisions even when you are unsure. School, marriage, therapy, jobs, whatever. • Statistics provides an approach to decision making under uncertainty. Sort of decision making by choosing the same way you would bet. Maximize expected utility (subjective value). • Comes from agronomy, where they were trying to decide what strain to plant.
- 30. 30 Statistical Hypotheses • While attempting to make decisions, some necessary assumptions or guesses about the populations or statements about the probability distribution of the populations made are called statistical hypothesis. These assumptions are to be proved or disproved • A predictive statement usually put in the form of a null hypothesis and alternate hypothesis Hypothesis Testing • Researcher bets in advance of his experiment that the results will agree with his theory and cannot be accounted for by the chance variation involved in sampling • Procedures which enable researcher to decide whether to accept or reject hypothesis or whether observed samples differ significantly from expected results
- 31. 31 Statistical Hypotheses • Statements about characteristics of populations, denoted H: -H: normal distribution, -H: N(28,13) • The hypothesis actually tested is called the null hypothesis, H0 -E.g., • The other hypothesis, assumed true if the null is false, is the alternative hypothesis, H1 -E.g., 13;28 == sm 100:0 =mH 100:1 ¹mH
- 32. 32 Testing Statistical Hypotheses - steps • State the null and alternative hypotheses • Assume whatever is required to specify the sampling distribution of the statistic (e.g., SD, normal distribution, etc.) • Find rejection region of sampling distribution –that place which is not likely if null is true • Collect sample data. Find whether statistic falls inside or outside the rejection region. If statistic falls in the rejection region, result is said to be statistically significant.
- 33. 33 The level of significance (a ) • a is known as the nominal level of significance. • If p-value < a, then we reject the null hypothesis in favor of the alternative hypothesis. • P-value is also known as the observed level of significance. • a needs to be pre-determined. (Usually 5%)
- 34. 34 Type I and Type II errors • Type I error is committed when a true null hypothesis is rejected. • a is the probability of committing type I error. • Type II error is committed when a false null hypothesis is not rejected. • b is the probability of committing type II error.
- 35. 35 Power of a test • The power of a test is the probability that a false null hypothesis is rejected. • Power = 1 - b, where b is the probability of committing type II error. • More powerful tests are preferred. At the design stage one should identify the desired level of power in the given situation.
- 36. 36 Decisions Null true Null False Accept Null Right Beta (type II error) Reject Null Alpha (type I error) Correct rejection (power) Population Condition Sample Decision No fire Fire Alarm silent Right, but… Beta Alarm on Alpha Correct rejection Three named probabilities: Alpha, beta, and power.
- 37. 37 Power of a test (1-β): • Value (1-β) indicates how well the test is working, i.e., value nearer to 1 means working well (test is rejecting Ho when it is not true) and value nearer to 0 means poorly working (not rejecting Ho when it is not true) • It indicates how well given test will enable us to minimize the probability of type II error (β), i.e., avoid making wrong decisions. Hypothesis testing cannot be foolproof. Sometimes test does not reject a Ho which is false (type II error). We would like β to be as small as possible or (1-β) to be as large as possible. • Operating Characteristic Function (L) L = 1 -H : Shows conditional probability of accepting Ho for all values of population parameters for a given sample size, whether or not the decision happens to be a correct one • OC curve -graphs showing the probabilities of type II error (β) under various hypotheses
- 38. 38 Factors influencing the Power • The power of a test is influenced by the magnitude of the difference between the null hypothesis and the true parameter. • The power of a test could be improved by increasing the sample size. • The power of a test could be improved by increasing a. (this is a very artificial way)
- 39. 39 One Tail or Two Tails The rejection region can fit into 1 or 2 tails of the sampling distribution of means. The RR is determined by the alternative hypothesis. Two Tails One Tail valueH =m:0 valueH a ¹m: valueH =m:0 orvalueH a >m: valueH a <m:
- 40. 40 Tails illustrated valueH a ¹m: 3210-1-2-3 ZZZ 1.96-1.96 Don't reject RejectReject Likely Outcome If Null is True 3210-1-2-3 ZZ Don't reject Reject Likely Outcome If Null is True 1.65 Two tails. One tail. valueH a >m: Note 1.96 vs. 1.65
- 41. 41 Example of 2 tails • Suppose: • Then: 75:;75:0 ¹= mm aHH 25,10 == Ns 92.7808.71 25 10 96.175 «=± X 78.9271.08 Don't reject RejectReject Likely Outcome If Null is True 75 Note 5 percent is split into two tails.
- 42. 42 Example of 1 tail • Suppose: 75:;75:0 >= mm aHH 25,10 == Ns 3.78 25 10 65.175 =+ 8079787776757473727170 X 0.20 0.16 0.12 0.08 0.04 0.00 8079787776757473727170 X Sampling Distribution of Means Likely Outcome if Null is True Don't Reject Reject 78.3 Note all 5 percent is at the top tail.
- 43. 43 Parametric or Standard Tests • Require measurements equivalent to at least an interval scale • Assume certain properties of parent population like -i) observations are from a normal population -ii) large random sample -iii) population parameters like mean, variance, etc. must hold good • Situations where above assumptions are not possible, non-parametric tests are used; As there is no model, these tests are also called distribution-free tests
- 44. 44 Parametric tests Z-test • Based on the normal probability distribution and even binomial in case of large samples. • For testing mean, variance, two individual samples, median, mode, correlation, coefficients etc. T-test • It is based on t-distribution and only incase of small samples • Used for testing difference between means of two samples, coefficient of simple & partial correlations, etc.
- 45. 45 (cont.) F-test • Used in the context of ANOVA and for the testing the significance of multiple correlation coefficients, comparing the variance of two independent samples, • Χ2 Test • Based on Chi-square distribution • Used for comparing a sample variance to a theoretical population variance
- 46. 46 Some Important Parametric Tests
- 47. 47 (cont.)
- 49. 49 The t-test Inferences about Population Means
- 50. 50 The t Distribution • We use t when the population variance is unknown (the usual case) and sample size is small (N<100, the usual case). • The t distribution is a short, fat relative of the normal. The shape of t depends on its df. As N becomes infinitely large, t becomes normal.
- 51. 51 Assumptions • The t-test is based on assumptions of normality • Two groups are independent • Homogeneity of variance -> can be tested by using F-test. • As long as the samples in each group are large and nearly equal, the t-test is robust, that is, still good, even though assumptions are not met.
- 52. 52 Normality Assumption • We assume normal distributions to figure sampling distributions and thus p levels. • Violations of normality have implications for testing means. Need to use non-parametric statistics or use data transformation • Can test for normality by using Kolmogrov-Simirnov test
- 53. 53 The F Distribution (1) • The F distribution is the ratio of two variance estimates: • Also the ratio of two chi-squares, each divided by its degrees of freedom: 2 2 2 1 2 2 2 1 . . s s est est s s F == 2 2 ( 1 2 )( /) / 2 1 v v F v v c c = In our applications, v2 will be larger than v1 and v2 will be larger than 2. In such a case, the mean of the F distribution (expected value) is v2 /(v2 -2).
- 54. 54 Testing Hypotheses about 2 Variances • Suppose -Note 1-tailed. • We find • Then df1=df2 = 15, and 2 2 2 11 2 2 2 10 :;: ssss >£ HH 7.1;16;8.5;16 2 22 2 11 ==== sNsN 41.3 7.1 8.5 2 2 2 1 === s s F Going to the F table with 15 and 15 df, we find that for alpha = .05 (1-tailed), the critical value is 2.40. Therefore the result is significant.
- 55. 55 Application of F Distribution • The F distribution is used in many statistical tests -Test for equality of variances. -Tests for differences in means in ANOVA. -Tests for regression models (slopes relating one continuous variable to another like SAT and GPA).
- 56. 56 Reference • Cohen, Louis and Manion, Lawrence. Research methods in education. London: Routledge, 1980. • Goode, William J and Hatt, Paul K. Methods on social research. London; Mc GrawHill, 1981. • 10.Gopal, M.H. An introduction to research procedures in social sciences. Bombay: Asia Publishing House, 1970. • Koosis, Donald J. Business statistics. New York: John Wiley,1972.