Statistics 226, Winter 1997

Midterm Exam

Outline of solutions

Notes:

A series of numbers is given in brackets for each problem; these are the point values assigned to the different parts of the problem. Thus, 1. [8/4/8] means that problem 1 was assigned 20 points: 8 for part (a), 4 for part (b) and 8 for part (c).

This is not meant to be a full solution set, but rather an indication of some of the important aspects of the problems, or an indication of the direction for a solution.

1. [8/4/8]

   a) Several answers.  The most appropriate might be the chi-squared test for 
   independence and logistic regression, although one could also compare mean 
   Karnofsky scores between groups, or conduct a linear trend test.  Measures 
   of association do not directly test the hypothesis, although they may still 
   be useful descriptors.
       
   b) Tests which take the ordering of the Karnofsky scores into account will 
   be more powerful, and also may lead to interpretible parameters.
   
   c) Under the hypothesis of 1:3 randomization, the expected number 
   randomized to Placebo is 53/4 = 13.25 and to MGDF 3*(53/4) = 39.75.  Using 
   the chi-squared goodness-of-fit test, we obtain
   
       chi-sq = (13-13.25)^2/13.25 +  (40-39.75)^/39.75 = 0.005

   with one degree of freedom.  The p-value is nearly one; the actual 
   allocations are about as close to the 1:3 ratio as they could be.
2. [4/4/4/4/4]

   a) 179/648 = 0.296
       
   b)  72/119 = 0.605
      110/306 = 0.359
       44/223 = 0.197
   
   c) 226/648 = 0.349
   
   d) Independence of males' education level and their opinions about the
      role of women.
      
   e) The chi-squared statistic of 57.09 on 2 df (calculated by Stata) 
      indicates a strong dependence between the variables.  The answer to 
      (b) above indicates that more inclusive views on women's roles are 
      associated with higher levels of education among men.
      
3. [6/6/6]  (+5 bonus)
      Note: A general description of the different sampling schemes does 
      not related to Table 1-m, and consequently did not get full credit.

   a) Men were sampled for a fixed period of time; this resulted in 
      completed surveys from 648 men, who were then cross-clasified by 
      education and opinion.
       
   b) Men are sampled until 119 people with grade-school education, 306 
      with high-school education, and 223 with some college are 
      obtained.  (These numbers are specified in advance by the survey 
      director.)  They are then classified by opinion within each education 
      level.
      
   c) Men are sampled until exactly 648 men are selected (this was the 
	  number which the survey budget would support).  They were then 
	  cross-clasified by education and opinion.
      
   d) Men are sampled until 119 people with grade-school education, 306 
      with high-school education, and 223 with some college are 
      obtained.  (These numbers are specified in advance by the survey 
      director.)  These men are placed in a room with 226 cards that say 
      "Agree" and 422 cards that say "Disagree".  The group is instructed 
      that each man must take one card, and that so far as possible, the 
      cards should reflect the individuals' opinions about the question.  
      The men are allowed to talk among themselves and to trade cards.  The 
      number of men from each education level who selected each type of 
      card is recorded in the table.
   
4. [6/6/9]

   a) The male-female odds ratio for white victims for a "firearm" attack is 
      1.08, which represents only a small association between sex and type 
      of attack.  For those who prefer statistical guidance, the 
      chi-squared statistic is 0.85 on 1 df (p=0.4).  The relationship, if 
      any, is very weak.
      
   b) The male-female odds ratio for black victims for a "firearm" attack is 
	  1.21, which represents a modest association between sex and type of 
	  attack.  The chi-squared statistic is 6.72 on 1 df (p=0.01).  This 
	  suggests that there is a real relationship (with men being more likely 
	  than women to be assaulted using firearms/explosives), even though it is 
	  not a large effect.
      
   c) After adjusting for racial differences, female victims are less likely 
      than male victims to be assualted with firearms/explosives (coeff = 
      -0.13).  After adjusting for any differences associated with the sex 
      of the victim, black victims are less likely than white victims to 
      be assaulted with firearms/explosives (coeff = -0.27).  There is 
      strong evidence that both of these effects are real (small 
      p-values).  A numerical interpretation would be that relative to 
      black women, assualts on males have odds increased by 15% and 
      assaults on whites have odds increased by an (additional) 31% for a 
      firearm attack.   [Note exp(0.273)=1.31 and exp(0.13)=1.15]
5. [8/7/6]

   a) 2 x [ -147.674 - ( -336.886 ) ] = 2 x 189.2 = 378.42
      df = 2 (1 df each for A, B).
       
   b) 147.674 - 145.212 = 2.46
      2 x 2.46 = 4.92.  This is a 1 df comparison.  The p-value is somewhat 
      less than 0.05, which suggests that C is helpful in any model which 
      contains A and B.
      
   c) If the regression coefficients in the logistic model are 6, 2, and 
	  1.5, then then odds factors corresponding to the conditions are 403 
	  for being in a single-parent household, 7.4 for overcrowded 
	  households, and 4.5 for foreign birth.  The amount by which the odds 
	  are increased for any individual child is the product of the odds 
	  factors which pertain to the child's classification.