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Intermediate Applied Statistics STAT 460

Intermediate Applied Statistics STAT 460. Lecture 20, 11/19/2004 Instructor: Aleksandra (Seša) Slavković sesa@stat.psu.edu TA: Wang Yu wangyu@stat.psu.edu. Revised schedule. Last lecture. Categorical Data. This lecture . Categorical Data/Response (ch. 18,19,20) Odds.

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Intermediate Applied Statistics STAT 460

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  1. Intermediate Applied Statistics STAT 460 Lecture 20, 11/19/2004 Instructor: Aleksandra (Seša) Slavković sesa@stat.psu.edu TA: Wang Yu wangyu@stat.psu.edu

  2. Revised schedule

  3. Last lecture • Categorical Data

  4. This lecture • Categorical Data/Response (ch. 18,19,20) • Odds

  5. Review: Categorical Variable • Notation: • Population proportion =  = sometimes we use p • Population size = N • Sample proportion = = X/n = # with trait / total # • Sample size = n • The Rule for Sample Proportions • If numerous samples of size n are taken, the frequency curve of the sample proportions ( ‘s) from the various samples will be approximately normal with the mean  and standard deviation • ~ N( , (1- )/n )

  6. One-sample approximate z test and z-interval for π.

  7. These tests can be extended to test the difference in parameters π between two groups.

  8. Difference between proportions These tests can be extended to test the difference in parameters π between two groups.

  9. Warning: z-tests for proportions are based on an approximation. They don’t work for small samples. It is often said that n is large enough if Because of improved computing power, an exact test based on the binomial distribution rather than the normal is now available in most software.

  10. Analysis Grid (ref. Handout)

  11. Contingency Table • A statistical tool for summarizing and displaying results for categorical variables • A two-way table if for two categorical variables • 2x2 Table, for two categorical variables, each with two categories • Place the counts of each combination of the two variables in the appropriate cells of the table. • Exploratory variable as labels for the rows, response variable as labels for the columns.

  12. Example • A university offers only two degree programs: English and Computer Science. Admission is competitive and there is a suspicion of discrimination against women in the admission process. Here is a two-way table of all applicants by sex and admission status: • These data show an association between the sex of the applicants and their success in obtaining admission.

  13. Marginal & Conditional Distributions • Marginal Distributions: • Exploratory Variable: add up values for the rows; take away response variable • In our example distribution is: 55, 85, 140 • Observed proportions: • ‘admit’ = 55/140 = 0.39 • ‘deny’ = 85/140 = 0.61 • NOTE: they add up to 1 • Response Variable: add up values for the columns; take away exploratory variable • In our example distribution is? • Observed proportions are: • Do they add up to 1?

  14. Marginal & Conditional Distributions • Conditional Distribution: • Conditional percentages; what percent of a particular row or a column a count in a cell is. • Conditional distribution of gender for those admitted: • % of admitted who are male = 35/55 = 0.63 = 63% • % of admitted who are female = ? • What is: • % of male applicants admitted = ? • % of female applicants admitted = ?

  15. Statistical Significance • An observed relationship is statistically significant if the chances of observing the relationship in the sample when there is no actual relationship in the population are small (usually less than 5%) • In other words, a relationship is statistically significant if that relationship is stronger than 95% of the relationships we would expect to see just by chance. • If we say that there was no statistically significant relationship found, that does not mean that there is no relationship at all! • Warnings: • If a sample size is small, strong relationships may not achieve significance • If a sample size is large, even minor relationships could achieve significance but these might not then have practical importance

  16. Chi-Squared Test (2 Test) • A Chi-Squared Test for independence • The Chi-Squared Statistics (2 ) for contingency table. • Follows 2 distribution • Skewed to the right • Min = 0, Max = infinity • As the strength of observed relationship in the sample increase, the statistic increases. • It combines info about a strength of the relationship and the sample size into a one number • Can be calculated for any size contingency table • For 2 x 2 table: if 2 > 3.84 then we have a statistically significant relationship • We either show (2 > 3.84) or fail to show significant relationship (if 2 < 3.8); we either reject (2 > 3.84 ) or fail to reject (2 < 3.84) the claim of independence between two variables that is our null hypothesis. • H0: variables are independent HA: variabls are NOT independent

  17. 2 • The chi-squared distribution with k-1 degrees of freedom acts as though it was the sum the squares of k-1 independent Normal(0,1) distributions. (Not that you need to know.) • See table on pages 1100-1101 in textbook.

  18. You Must Know: • How to calculate 2 statistic • Compute the expected numbers • Compare the expected and observed numbers • Compute the 2 statistic • How to compare it to 3.84 for 2x2 tables • How to make proper conclusion about statistical relationship and in general about the question of interest for any two-way and k-way tables.

  19. For our example: • Computing 2statistic: • Expected number = the number of counts (individuals) that we expect to fall in a particular cell = (row total)(column total)/(table total) • Expected number of admitted male students = (55 x 80)/140 = 31.42 • Expected number of admitted female students = ? • Observed number = the number of counts in the cell • Observed number of admitted male students = 35 • Observed number of admitted female students = ? • Compare the observed and expected number : ( observed – expected)2/(expected number) For male students: (35 - 31.42)2/(31.42) = 0.41 For female students: = ? • Compute the statistic = Sum all the above calculated numbers for all the cells • In our case 2= 1.58 • Compare it to 3.84 • Is it statistically significant? Are admission decisions independent of the gender?

  20. Relative Risk, Increased Risk, Odds Ratio • Quantifications of the chances of a particular outcome and how do these chances change • What are the chances that a randomly selected individual would fall into a particular category for a categorical variable. • There are two basic ways to express these chances: • Proportions = expressing one category as a proportion of the total • Proportion of admitted students who are female = 20/55 = 0.36 • Odds = comparing one category to another • Odds of being admitted = 55 to 85 = 55/85 to 1

  21. Expressing Proportions & Odds • There are 4 equivalent ways to express proportions: • Percent = Proportion = Probability = Risk • 36% (percent) of all admitted students are females • The proportion of females admitted is 0.36 • The probability that a female would be admitted is 0.36 • The risk for a female to be admitted is 0.36 • Odds = expressed by reducing the numbers with and without a characteristic we are interested in to the smallest possible whole number: • The odds of being admitted = 55 to 85 = 7 to 11 = 7/11 to 1 • Going back and forth between proportions and odds: • If the proportion has value p then the odds are: /(1- )to 1 • If the odds of having a characteristic are a to b, then the proportion with the characteristic is a/(a+b)

  22. Generalized forms for the expressions: • Percentage with the characteristic = (number with the characteristic/total) x 100% • Proportion with the characteristic = (number with the characteristic/total) • Probability of having the characteristics = (number with the characteristic/total) • Risk of having the characteristic = (number with the characteristic/total) • Odds of having the characteristic = (number with the characteristic/number without characteristics) to 1 • = /(1- )

  23. Types of Risk: Relative risk & Increased Risk • Relative risk = the ratio of the risks for each category of the exploratory variable • Relative risk of being a female based on whether you are rejected or accepted: • Risk for being rejected if you are female = 40/85 = 0.47 • Risk of being accepted if you are female = 20/55 = 0.36 • Relative risk = 0.47/0.36 = 1.31 to 1 • What does this mean? • What does a relative risk of 1 mean? • Increased Risk = usually, the percent increase in risk • Increased risk = (change in risk/original risk) x 100% • Change in risk = 0.47 – 0.36 = 0.11 • Original risk = Baseline risk = 0.36 • Increased risk = 0.11/0.36x 100% = 0.31 = 31% • There is a 23% increase in the chances of females to be rejected • Increased risk = (relative risk – 1.0) x 100% • Increased risk = (1.31 – 1.0) x 100% = 31%

  24. Odds Ratio • First calculate the odds of having a characteristic versus not having it: • Odds for female being admitted = 20/35 =0.571429 • Odds for female being rejected = 40/45= 0.888889 • Then take the ratio of these odds: • Odds ratio = 0.888889/ 0.571429 = 1.5556 • Not too close to 1.31, but sometimes it can be close to relative risk • Odds ratio = (upper left * lower right)/(upper right * lower left) • Sometimes you need to reverse denominator and numerator so that the ratio is greater than 1 (easier to interpret)

  25. Misleading items about Risk/Odds • The baseline risk is missing • The time period of the risk is not identified • The reported risk is not necessarily your risk (relative risk vs. your risk) • Retrospective vs. Prospective study • Prospective: take a random sample and record success and failure in the future • Retrospective: take a random sample and record success and failure that happened in the past • In retrospective study you can meaningfully interpret odds ratio, but not individual odds

  26. Simpson’s Paradox • Lurking variable = A variable that changes the nature of association even reverses direction of relationship between two other variables. • A nature of association changes due to a lurking variable • In our example we didn’t consider type of a program (major) as a variable. What happens if we do, and if construct two separate tables, one for each major?

  27. Example of Simpson’s Paradox • Computer Science admits each 50% of males and females • English takes ¼ of both males and females • Now there doesn’t seem to be an association between sex and admission decision in either program • Hence, type of program was a lurking variable

  28. Commands in SAS • To create contingency tables, calculate chi-square statistic, etc… • Statistics/Table Analysis • To run the logistic regression • Statistics/Regression/Logistic

  29. Next • Lab Monday • Categorical Data, • Logistic Regression -- we will work through the lab together and learn about logistic regression • Project II

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