25 biostat_lecture_3.pdf

Hypothesis  Research Hypothesis Research hypothesis is a definite statement that there is a relationship between populations regarding certain variable Examples: The Zika Virus Disease (ZVD) Education affects the Desire to Change Family Planning. Equivalently speaking,...

Hypothesis  Research Hypothesis Research hypothesis is a definite statement that there is a relationship between populations regarding certain variable Examples: The Zika Virus Disease (ZVD) Education affects the Desire to Change Family Planning. Equivalently speaking, “The proportions of women with the desire to change Family Planning are different between women with and without ZVD education.” There is a difference between genders regarding the number of times per week that ice cream is consumed. Direction of Research Hypotheses Nondirectional research hypotheses States that there is a difference (or relationship) between groups but doesn’t specify the direction Example: There is a difference between genders (men & women) regarding number of times per week that ice cream is consumed. Directional research hypotheses States that there is a difference (or relationship) between groups AND specifies direction Example: The average number of times that men eat ice cream per week is greater than the average number of times that women each ice cream per week. Statistical Way: Hypotheses Starting Point: Null hypothesis (H0 ) essentially states that there is no difference between populations regarding some variable. Examples: The proportions of women with the desire to change family planning are not different between women with and without ZVD education. There is no difference between genders regarding the number of times per week that ice cream is consumed. Why the Null Hypothesis (H0)? Starting point against which actual outcomes can be measured When we pose a research question, we want to know whether the outcome is due to the treatment (independent variable) or due to chance (in which case our treatment is probably not effective). For example, the claim that ZVD education increases the proportion of women with the desire to change family planning does not tell the exact amount of increase. It will be difficult to specify the probability of each of the possible increases that would support our research hypothesis. On the other hand, the null hypothesis is straightforward -- the women with and without ZVD education are from the same population (that the education has no effect)? There is only one set of statistical probabilities -- chance. Instead of directly testing the research claim (H1), we test H0. If we can reject H0, we can accept H1. To put it another way, the fate of the research hypothesis depends upon what happens to H0.  Research Hypothesis becomes the alternative hypothesis (Ha or H1 ) in a statistical test setting. Statistical Expression of hypotheses: let µ represent the parameter of interest, e.g. mean number of times per week that ice cream is consumed. µ1 and µ2 represent the means of men and women Null Hypothesis alternative Hypothesis (research hypothesis) H 0 : µ1 = µ2 vs H a : µ1 ≠ µ2 Two-tailed Or H 0 : µ1 = µ2 vs Ha : µ1 > µ2 One-tailed Or H 0 : µ1 = µ2 vs H a : µ1 < µ2 One-tailed Note: tailed and sided mean the same.  Hypothesis Testing A hypothesis test (or test of significance) is a standard procedure for testing a claim about a property of one or more populations. Rare Event Rule for Inferential Statistics If, under a given assumption, the probability of a particular observed event is exceptionally small, we conclude that the assumption is probably not correct. Example: Consider the claim that the mean weight of airline passengers is 180 lb in the summer (the current value used by the Federal Aviation Administration). Research hypothesis: the mean weight of airline passengers is more than 180 lb in the summer Example: 1) Express the given claim in symbolic form. The claim that the mean is 180 lb is expressed in symbolic form as µ = 180 lb. 2) If µ = 180 lb is false, let the alternative claim be µ > 180 lb Example: 3) Of the two symbolic expressions µ = 180 lb and µ > 180 lb, we see that µ > 180 lb does not contain equality, so we let the Null hypothesis H0 be µ = 180 lb. alternative hypothesis H1 be µ > 180 lb.  184 202 189 Use the data to test the hypothesis: 190 174 175 208 One sample of data: n= 30 196 193 193 180 158 171 Sample statistic: the sample mean 184 169 160 160 Sx 188 176 147 x = 154 n 167 196 182 167 173 173 187 162 185  Test Statistic The test statistic is a value used in making a decision about the null hypothesis, and is found by converting the sample statistic to a score with the assumption that the null hypothesis is true.  The standard error of mean (SEM) reflects the uncertainty in the mean estimated from a sample. If we know the standard deviation of the population, SEM= σ n Test statistic for mean P-Value The P-value (or p-value or probability value) is the probability of getting a value of the test statistic that is at least as extreme as the one representing the sample data, assuming that the null hypothesis is true. ? µ = 180 or z=0 Test Statistic z = 1.5  P-Value For the example, P-value = area to the right of the test statistic 0.067 µ = 180 or z=0 Test Statistic z = 1.5 Application of the rare event rule: Is this p-value exceptionally small? Statistical Significance: A test statistic and its corresponding sample statistic would rarely occur by chance under the Null hypothesis. Then, it is called statistically significant. Common choices for the significance level a are 0.05, 0.01, and 0.10.  Decision Criterion Popular P-value method: Using the significance level a, If P-value £ a , reject H0. If P-value > a , fail to reject H0.  Conclusions in Hypothesis Testing The initial conclusion will always be one of the following: 1. Reject the null hypothesis. 2. Fail to reject the null hypothesis. Example: At the 0.05 (or 5%) significance level, we fail to reject the null hypothesis, µ = 180 lb. Interpretation: There is not enough evidence that the mean weight of airline passengers is different from 180 lb in the summer. (p-value=0.067) Logics behind the statistical hypothesis testing: p-value method: Assuming the Null hypothesis is True Compute the sample statistics , and convert it to a test statistics z or t Compute the p-value: Assign the probability corresponding to the z (or t) statistics Compare the p-value to a pre-defined significance level 𝛼, and make decision P-value > 𝜶, not a rare event P-value ≤ 𝜶, a rare event Under the assumption of Null hypothesis Under the assumption of Null hypothesis Nothing wrong with the procedure; Nothing wrong with the procedure; No evidence that Null hypothesis must not be True Enough evidence that Null hypothesis must not be True Fail to reject the Null Hypothesis Reject the Null Hypothesis Errors of the decisions based on statistical hypothesis testing: Type I Error A Type I error is the mistake of rejecting the null hypothesis when it is actually true. Equivalently, a Type I error is the mistake of rejecting the null hypothesis when the alternative hypothesis is false. Type II Error A Type II error is the mistake of failing to reject the null hypothesis when it is actually false. Equivalently, a Type II error is the mistake of failing to reject the null hypothesis when the alternative hypothesis is true. Type I and Type II Errors Confidence Interval Estimating the population Mean with the sample meanx : Point estimation, i.e., it takes a single value each time; There is uncertainty. The sample mean approximates a normal distribution as the sample size increases regardless of the distribution of the population: Center around the same population mean Variation of the sample mean depends on the sample size: As n increases, the distribution of sample mean x has smaller spread. In other words, the uncertainty with the sample mean x decreases as the sample size increases.  x (blood pressure) µ z=0 Z-score Interval estimator: It gives us a range of values with certain probability to contain the population mean. The interval is called a confidence interval. It tells us the probability for the interval to contain the population mean. That is confidence level. A range of values used to estimate the true mean value of a population with a specified probability Create a confidence interval for a mean when the population standard deviation σ is known. The choice of the z value is determined by the confidence level specified. x - Z * SEM x x + Z * SEM Where, SEM= σ n  x - Z * SEM x x + Z * SEM e.g., a 95% CI: 120 – 6 x 120 + 6 114 x 126 [114, 126] Interpretation: e.g. 95% confidence interval If repeated samples were taken and the 95% confidence interval computed for each sample, 95% of the intervals would contain the population mean. In other words, we are 95% confident that the interval would contain the true population mean. x - Z * SEM x x + Z * SEM The width of confidence interval 1)It increases as the confidence level increases. 2)It decreases as the sample size n increases.  Margin of Error A confidence interval is typically expressed as: sample mean ± m – m is called the margin of error e.g. 95% CI for the systolic blood pressure sample mean ± m = 120 ± 6 Margin of Error For example, when they say “a poll found 52% of people approve of the president” and you see on the screen “margin of error: 2%”. Then you know they are talking about a confidence interval of (50%, 54%). (they tend not to give the confidence level as that is a bit technical for a general news broadcast). Correlation A correlation exists between two variables when the values of one are somehow associated with the values of the other in some way. The linear correlation coefficient r, also known as Pearson Correlation Coefficient, measures the strength of the linear relationship between the paired quantitative x- and y-values in a sample.  Exploring the Data We can often see a relationship between two variables by constructing a scatterplot. Example from a medical publication:  Exploring the Data Figures below following shows scatterplots with different characteristics. Scatterplots of Paired Data Scatterplots of Paired Data  Properties of the Pearson Correlation Coefficient r 1. –1 £ r £ 1; the absolute value of r indicates the strength of relationship while the sign describes direction (positive or negative). 2. if all values of either variable are converted to a different scale, the value of r does not change. 3. The value of r is not affected by the choice of x and y. Interchange all x- and y-values and the value of r will not change. 4. r is very sensitive to outliers, they can dramatically affect its value. Strength of the Pearson Correlation Coefficient r Size of the Correlation Coefficient General Interpretation.8 to 1.0 Very strong relationship.6 to.8 Strong relationship.4 to.6 Moderate relationship.2 to.4 Weak relationship.0 to.2 Weak or no relationship  Common Errors Involving Correlation 1. Causation: It is wrong to conclude that correlation implies causality. 2. Linearity: There may be some relationship between x and y even when there is no linear correlation. Scatterplots of Paired Data Comments on the exam questions: There will be no calculations. The exam questions focus on the comprehension of statistical concepts and principles, and may Include the interpretations of graphs such as normal probability curve and histogram.