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Significance Tests

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  1. Significance Tests • Hypothesis - Statement Regarding a Characteristic of a Variable or set of variables. Corresponds to population(s) • Majority of registered voters favor health care reform • Average salary progressions differ for male executives whose spouses work than for those whose spouses “stay at home” • Significance Test - Means of using sample statistics (and their sampling distributions) to compare their observed values with hypothesized value of corresponding parameter(s)

  2. Elements of Significance Test (I) • Assumptions • Data Type: Quantitative vs. Qualitative • Population Distribution: Some methods assume normal • Sampling Plan: Simple Random Sampling • Sample Size: Some methods have sample size requirements for validity • Hypotheses • Null Hypothesis (H0): A statement that parameter(s) take on specific value(s) (Often: “No effect”) • Alternative Hypothesis (Ha): A statement contradicting the parameter value(s) in the null hypothesis

  3. Elements of Significance Test (II) • Test Statistic: Quantity based on the sample data to test the null hypothesis. Typically is based on a sample statistic, parameter value under H0 , and the standard error. • P-value (P): The probability that we would obtain a test statistic at least as contradictory to the null hypothesis as our computed test statistic, if the null hypothesis is true. • Small P-values mean the sample data are not consistent with the parameter value(s) under H0

  4. Elements of Significance Test (III) • Conclusion (Optional) • If the P-value is sufficiently small, we reject H0 in favor of Ha . The most widely accepted minimum level is 0.05, and the test is said to be significant at the .05 level. • If the P-value is not sufficiently small, we fail to reject (but not necessarily accept) the null hypothesis. • Process is analogous to American judicial system • H0: Defendant is innocent • Ha: Defendant is guilty

  5. Significance Test for Mean (Large-Sample) • Assumptions: Random sample with n  30, quantitative variable • Null Hypothesis: H0: m = m0 (typically no effect or change from standard) • Alternative Hypothesis: Ha: m  m0 (2-sided alternative includes both > and <) • Test Statistic: • P-value: P=2P(Z  |zobs|)

  6. Example - Mercury Levels • Population: Patients visiting private internal medicine clinic in S.F. (High-end fish consumers) • Variable: Mercury levels (microg/L) • Sample: 66 Females • Recommended maximum level: 5.0 microg/L • Null hypothesis: H0: m = 5.0 (Mean level=RML) • Alternative hypothesis: Ha: m 5.0 (Mean  RML) • Sample Data:

  7. Example - Mercury Levels • Test Statistic: • P-Value: • P=2P(Z  5.41) < 2P(Z  5.00) = 2(.000000287)= .000000574  0 • Conclusion: Very strong evidence that the population mean mercury level is above RML Source: Hightower and Moore (2003), “Mercury Levels in High-End Consumers of Fish, Environ Health Perspect, 111(4):A233

  8. Miscellaneous Comments • Effect of sample size on P-values: For a given observed sample mean and standard deviation, the larger the sample size, the larger the test statistic and smaller the P-value (as long as the sample mean does not equal m0) • Equivalence between 2-tailed tests and confidence intervals: If a (1-a)100% CI for m contains m0, the P-value will be larger than a • 1-sided tests: Sometimes researchers have a specific direction in mind for alternative hypothesis prior to collecting data.

  9. Example - Crime Rates (1960-80) • Sample: n=74 Chicago Neighborhoods • Goal: Show the average delinquency rate in the population of all such neighborhoods has increased from 1960-1980 • Variable: Y = DR1980-DR1960 • H0: m = 0 (No change from 1960-1980) • Ha: m > 0 (Higher in 1980, see Y above) • Sample Data:

  10. Example - Crime Rates (1960-80) • Test Statistic: • P-value: (Only interested in larger positive values since 1-sided) • Conclusion: Strong evidence that the true mean delinquency rate among all neighborhoods that this sample was taken from has increased from 1960 to 1980. Source: Bursik and Grasmick (1993), “Economic Deprivation and Neighborhood Crime Rates, 1960-1980”, Law & Society Review, Vol. 27, pp 263-284

  11. Significance Test for a Proportion (Large-Sample) • Assumptions: • Qualitative Variable • Random sample • Large sample: n 10/min(p0 , 1- p0) • Hypotheses: • Null hypothesis: H0: p= p0 • Alternative hypothesis: Ha: p p0 (2-sided) • Ha+ : p> p0 Ha- : p< p0 (1-sided, prior to data)

  12. Significance Test for a Proportion (Large-Sample) • Test statistic: • P-value: • Ha: p p0 P = 2P(Z  |zobs|) • Ha+ : p> p0 P = P(Z  zobs) • Ha- : p< p0 P = P(Z  zobs) • Conclusion: Similar to test for a mean

  13. Decisions in Tests • a-level (aka significance level): Pre-specified “hurdle” for which one rejects H0 if the P-value falls below it. (Typically .05 or .01) • Rejection Region: Values of the test statistic for which we reject the null hypothesis • For 2-sided tests with a = .05, we reject H0 if |zobs| 1.96

  14. Error Types • Type I Error: Reject H0 when it is true • Type II Error: Do not reject H0 when it is false

  15. Error Types • Probability of a Type I Error: a-Level (significance level) • Probability of a Type II Error: b - depends on the true level of the parameter (in the range of values under Ha ). • For a given sample size, and variability in data, the Type I and Type II error rates are inversely related • Conclusions wrt H0 are the same whether a hypothesis test or CI is conducted (fixed a)

  16. Miscellaneous Issues • Statistical vs Practical Significance: With very large sample sizes, we can often obtain very small P-values even when the sample quantity is very close to the parameter value under H0. Always consider the estimate as well as P-value. • While hypothesis tests and confidence intervals give similar conclusions wrt H0, the CI gives a credible set of parameter values, which can be more specific than test

  17. Small-sample Inference for m • t Distribution: • Population distribution for a variable is normal • Mean m, Standard Deviation s • The t statistic has a sampling distribution that is called the t distribution with (n-1) degrees of freedom: • Symmetric, bell-shaped around 0 (like standard normal, z distribution) • Indexed by “degrees of freedom”, as they increase the distribution approaches z • Have heavier tails (more probability beyond same values) as z • Table B gives tA where P(t > tA) = A for degrees of freedom 1-29 and various A

  18. Small-Sample 95% CI for m • Random sample from a normal population distribution: • t.025,n-1 is the critical value leaving an upper tail area of .025 in the t distribution with n-1 degrees of freedom • For n  30, use z.025 = 1.96 as an approximation for t.025,n-1

  19. t test for a mean • Assumptions: Random sample for a quantitative variable with a normal probability distribution • Hypotheses: • H0: m = m0Ha: m  m0 (2-sided) • Test Statistic: • P-Value: 2P(t > |tobs|) • Conclusions as before, as well as 1-sided tests