BMS 511 Introduction to Inference - Statistical Analysis PDF

BMS 511 Biostats & Statistical Analysis Chapter 14 Introduction to inference Guang Xu, PhD, MPH Assistant Professor of Biostatistics and Public Health College of Osteopathic Medicine Marian University...

BMS 511 Biostats & Statistical Analysis Chapter 14 Introduction to inference Guang Xu, PhD, MPH Assistant Professor of Biostatistics and Public Health College of Osteopathic Medicine Marian University Copyright © 2018 W. H. Freeman and Company  Previous Learning Objectives Sampling distributions Parameter versus statistic Sampling distributions Sampling distribution of the sample mean The central limit theorem Sampling distribution of the sample proportion The law of large numbers Copyright © 2018 W. H. Freeman and Company  Learning Objectives Demonstrate the following features of inference Statistical estimation Margin of error and confidence levels Confidence interval for a Normal population mean (σ known) Hypothesis testing The P-value Test for a Normal population mean (σ known) Tests from confidence intervals Copyright © 2018 W. H. Freeman and Company  Statistical estimation If you picked different samples from a population, you would probably get different sample means () and virtually none of them would actually equal the true population mean, . Copyright © 2018 W. H. Freeman and Company  Use of sampling distributions If the population is N(μ, σ), the sampling distribution is N(μ, σ/√n). If not, the sampling distribution is ~N(μ, σ/√n) if n is large enough.  We take one random sample of size n, and rely on the known properties of the sampling distribution. Copyright © 2018 W. H. Freeman and Company  Visualization of confidence When we take a random sample using a computer applet, we can compute the sample mean and an interval of size plus-or-minus 2 σ/√n around the mean. Based on the 68–95–99.7% rule, we can expect that about 95% of all intervals computed with this method capture the parameter μ. Copyright © 2018 W. H. Freeman and Company  Confidence intervals A confidence interval is a range of values with an associated probability, or confidence level, C. This probability quantifies the chance that the interval contains the unknown population parameter. We have confidence C that µ falls within the interval computed. Copyright © 2018 W. H. Freeman and Company  The margin of error A confidence interval (“CI”) can be expressed as: – a center ± a margin of error m: µ within m – an interval: µ within (m) to (m) The confidence level C (in %) represents an area of corresponding size C under the sampling distribution. Copyright © 2018 W. H. Freeman and Company  Margin of error interpretation (1 of 2) A 95% confidence interval for the mean body temperature (in °F) was computed as (98.1, 98.4), based on body temperatures from a sample of 130 healthy adults. The correct interpretation of this interval is: A. 95% of observations in the sample have body temperature between 98.1 and 98.4°F. B. 95% of the individuals in the population should have body temperature between 98.1 and 98.4°F. C. We are 95% confident that the population mean body temperature is a value between 98.1 and 98.4°F. Copyright © 2018 W. H. Freeman and Company  Margin of error interpretation (2 of 2) A 95% confidence interval for the mean body temperature (in °F) was computed as (98.1, 98.4), based on body temperatures from a sample of 130 healthy adults. The margin of error for this interval is: A. 95% B. 98.25°F C. 0.3°F D. 0.15°F. Copyright © 2018 W. H. Freeman and Company CI for a Normal population mean (σ known) When taking a random sample from a Normal population with known standard deviation σ, a level C confidence interval for µ is: or σ/√n is the standard deviation of the sampling distribution C is the area under the N(0,1) between −z* and z* Copyright © 2018 W. H. Freeman and Company  Finding z-values We can use a table of z- and t-values (Table C). For a given confidence level C, the appropriate z*-value is listed in the same column. Copyright © 2018 W. H. Freeman and Company  Confidence interval calculations (1 of 2) Density of bacteria in solution Measurement equipment has Normal distribution with standard deviation σ = 1 million bacteria/mL of fluid. Three measurements made: 24, 29, and 31 million bacteria/mL. Mean: 28 million bacteria/ml. Find the 99% and 90% CI. 99% confidence interval for the true density, z* = 2. 576 Copyright © 2018 W. H. Freeman and Company  Confidence interval calculations (2 of 2) 90% confidence interval for the true density, Confidence level C 90% 95% 99% Critical value z* 1.645 1.960 2.576 Copyright © 2018 W. H. Freeman and Company  Confidence level and margin of error (1 of 2) The confidence level C determines the value of z* (in Table C). The margin of error also depends on z*. Copyright © 2018 W. H. Freeman and Company  Confidence level and margin of error (2 of 2) Higher confidence C implies a larger margin of error m (less precision more accuracy). A lower confidence level C produces a smaller margin of error m (more precision less accuracy).  We want high confidence, but also narrow intervals. We generally choose the confidence level first, then increase n to achieve a narrow interval. Copyright © 2018 W. H. Freeman and Company  Hypothesis testing (1 of 2) Someone makes a claim about the unknown value of a population parameter. We check whether or not this claim makes sense in light of the “evidence” gathered (sample data). A test of statistical significance tests a specific hypothesis using sample data to decide on the validity of the hypothesis. Copyright © 2018 W. H. Freeman and Company  Hypothesis testing (2 of 2) Blood levels of inorganic phosphorus are known to vary Normally among adults, with mean 1.2 and standard deviation 0.1 mmol/L. But is the true mean inorganic phosphorus level lower among the elderly? The average inorganic phosphorus level of a random sample of 12 healthy elderly subjects is 1.128 mmol/L. Is this smaller mean phosphorus level simply due to chance variation? Is it evidence that the true mean phosphorus level in elderly individuals is lower than 1.2 mmol/L? Copyright © 2018 W. H. Freeman and Company  Null and alternative hypotheses The null hypothesis, H0, is a very specific statement about a parameter of the population(s). The alternative hypothesis, Ha, is a more general statement that complements yet is mutually exclusive with the null hypothesis. Phosphorus levels in the elderly: H0: µ = 1.2 mmol/L Ha: µ < 1.2 mmol/L (µ is smaller due to changing physiology) Copyright © 2018 W. H. Freeman and Company  One-sided versus two-sided alternatives A two-tail or two-sided alternative is symmetric: Ha: µ  [a specific value or another parameter] A one-tail or one-sided alternative is asymmetric and specific: Ha: µ < [a specific value or another parameter] OR Ha: µ > [a specific value or another parameter] What determines the choice of a one-sided versus two-sided test is the question we are asking and what we know about the problem before performing the test. If the question or problem is asymmetric, then Ha should be one-sided. If not, Ha should be two-sided. Copyright © 2018 W. H. Freeman and Company  Examples of hypotheses (1 of 2) Is the active ingredient concentration as stated on the label (325 mg/tablet)? H0: µ = 325 Ha: µ ≠ 325 Is nicotine content greater than the written 1 mg/cigarette, on average? H0: µ = 1 Ha: µ > 1 Does a drug create a change in blood pressure, on average? H0: µ = 0 Ha: µ ≠ 0 Copyright © 2018 W. H. Freeman and Company  Examples of hypotheses (2 of 2) Does a particular stream have an unhealthy mean oxygen content (a level below 5 mg per liter)? Ecologists collect a liter of water from each of 45 random locations along a stream and measure the amount of dissolved oxygen in each. They find a mean of 4.62 mg per liter. H0:  = 5 Ha:  < 5 Copyright © 2018 W. H. Freeman and Company  The P-value Phosphorus levels vary Normally with standard deviation s = 0.1 mmol/L. H0: µ = 1.2 mmol/L versus Ha: µ < 1.2 mmol/L The mean phosphorus level from the 12 elderly subjects is 1.128 mmol/L. What is the probability of drawing a random sample with a mean as small as this one or even smaller, if H0 is true? P-value: The probability, if H0 was true, of obtaining a sample statistic at least as extreme (in the direction of Ha) as the one obtained. Copyright © 2018 W. H. Freeman and Company  Visualizing the P-value Phosphorus levels vary Normally with standard deviation s = 0.1 mmol/L. H0: µ = 1.2 mmol/L versus Ha: µ < 1.2 mmol/L The mean phosphorus level The chance of seeing a from the 12 elderly subjects sample mean less than our is 1.128 mmol/L. observed average of 1.128 is 0.0063. Copyright © 2018 W. H. Freeman and Company  Interpreting a P-value Could random variation alone account for the difference between H0 and observations from a random sample?  Small P-values are strong evidence AGAINST H0 and we reject H0. The findings are “statistically significant.”  P-values that are not small don’t give enough evidence against H0 and we fail to reject H0. Beware: We can never “prove H0.” Copyright © 2018 W. H. Freeman and Company  Range of P-values P-values are probabilities, so they are always a number between 0 and 1. The order of magnitude of the P-value matters more than its exact numerical value. Copyright © 2018 W. H. Freeman and Company  The significance level α (1 of 2) The significance level, α, is the largest P-value tolerated for rejecting H0 (how much evidence against H0 we require). This value is decided arbitrarily before conducting the test. – When P-value ≤ α, we reject H0. – When P-value > α, we fail to reject H0. Copyright © 2018 W. H. Freeman and Company  The significance level α (2 of 2) Example: Industry standards require a significance level α of 5%. Does the packaging machine need revision? A two-sided test is performed on a sample of data. The P-value is 4.56%. Because the P-value < 5%, the results are statistically significant at significance level 0.05. Copyright © 2018 W. H. Freeman and Company  Test for a population mean (σ known) To test H0: µ = µ0 using a random sample of size n from a Normal population with known standard deviation σ, we use the null sampling distribution N(µ0, σ√n). The P-value is the area under N(µ0, σ√n) for values of x̅ at least as extreme in the direction of Ha as that of our random sample. Calculate the z-value then use Table B or C. Or use technology. Copyright © 2018 W. H. Freeman and Company  One-sided versus two-sided P-values (1 of 2) One-sided (one-tailed) test Two-sided (two-tailed) test Copyright © 2018 W. H. Freeman and Company  One-sided versus two-sided P-values (2 of 2) To calculate the P-value for a two-sided test, use the symmetry of the normal curve. Find the P-value for a one- sided test and double it. Copyright © 2018 W. H. Freeman and Company P-values in Table C Copyright © 2018 W. H. Freeman and Company  Hypothesis testing example (1 of 2) Do the elderly have a mean phosphorus level below 1.2 mmol/L? H0: µ = 1.2 versus Ha: µ < 1.2 What would be the probability of drawing a random sample such as this or worse if H0 was true? Copyright © 2018 W. H. Freeman and Company  Hypothesis testing example (2 of 2) Table B: P-value = P(z ≤ –2.49) = 0.0064 Table C: one-sided P-value is between 0.005 and 0.01 (use |z|). The probability of getting a random sample average so different from µ0 is so low that we reject H0. Conclusion: The mean phosphorus level among the elderly is significantly less than 1.2 mmol/L. Copyright © 2018 W. H. Freeman and Company  Tests from confidence intervals Because a two-sided test is symmetric, you can easily use a confidence interval to test a two-sided hypothesis. In a two-sided test, C = 1 – α. C confidence level α significance level Copyright © 2018 W. H. Freeman and Company Logic of confidence interval test (1 of 2) We found a 99% confidence interval for the true bacterial density of , or 26.5 to 29.5 million bacteria/ per mL. With 99% confidence, could the population mean be µ =25 million/mL? µ = 29? Copyright © 2018 W. H. Freeman and Company  Logic of confidence interval test (2 of 2) A confidence interval gives a black and white answer: Reject or don’t reject H0. But it also estimates a range of likely values for the true population mean µ. A P-value quantifies how strong the evidence is against the H0. But if you reject H0, it doesn’t provide any information about the true population mean µ. Copyright © 2018 W. H. Freeman and Company  Learning Objectives Demonstrate the following features of inference Statistical estimation Margin of error and confidence levels Confidence interval for a Normal population mean (σ known) Hypothesis testing The P-value Test for a Normal population mean (σ known) Tests from confidence intervals Copyright © 2018 W. H. Freeman and Company