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Solution Manual for Fundamentals of Biostatistics, 8th Edition

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    Complete Solutions Manual to Accompany Fundamentals of Biostatistics EIGHTH EDITION Bernard Rosner Harvard University, Cambridge, MA Prepared by Roland A. Matsouaka Duke University, Durham, NC

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    Contents Chapter 2 Descriptive Statistics ....................................................................................................... 2 Chapter 3 Probability ..................................................................................................................... 21 Chapter 4 Discrete Probability Distributions ................................................................................. 43 Chapter 5 Continuous Probability Distributions ............................................................................ 65 Chapter 6 Estimation...................................................................................................................... 93 Chapter 7 Hypothesis Testing: One-Sample Inference ............................................................... 119 Chapter 8 Hypothesis Testing: Two-Sample Inference .............................................................. 146 Chapter 9 Nonparametric Methods .............................................................................................. 192 Chapter 10 Hypothesis Testing: Categorical Data ....................................................................... 216 Chapter 11 Regression and Correlation Methods ........................................................................ 267 Chapter 12 Multisample Inference ............................................................................................... 322 Chapter 13 Design and Analysis Techniques for Epidemiologic Studies .................................... 358 Chapter 14 Hypothesis Testing: Person-Time Data .................................................................... 413

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    2 D ESCRIPTIVE S TATISTICS 2.1 We have x x n i     215 25 8 6 . days median     n 1 2 th largest observation = 13th largest observation = 8 days 2.2 We have that s 2  x i  x   2 i  1 25  24  5  8.6   2    4  8.6   2 24  784 24  32.67 s   standard deviation = variance days 5 72 . range largest smallest o b servation days      30 3 27 2.3 Suppose we divide the patients according to whether or not they received antibiotics, and calculate the mean and standard deviation for each of the two subsamples: x s n Antibiotics 11.57 8.81 7 No antibiotics 7.44 3.70 18 Antibiotics - x 7 8.50 3.73 6 It appears that antibiotic users stay longer in the hospital. Note that when we remove observation 7, the two standard deviations are in substantial agreement, and the difference in the means is not that impressive anymore. This example shows that x and s 2 are not robust; that is, their values are easily affected by outliers, particularly in small samples. Therefore, we would not conclude that hospital stay is different for antibiotic users vs. non-antibiotic users.

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    CHAPTER 2/ DESCRIPTIVE STATISTICS 3 2.4-2.7 Changing the scale by a factor c will multiply each data value x i by c , changing it to cx i . Again the same individual’s value will be at the median and the same individual’s value will be at the mode, but these values will be multiplied by c . The geometric mean will be multiplied by c also, as can easily be shown: Geometric mean old geometric mean        [( )( ) ( )] ( ) ( ) / / / cx cx cx c x x x c x x x c n n n n n n n 1 2 1 1 2 1 1 2 1    The range will also be multiplied by c . For example, if c  2 we have: Original Scale –3 –2 –1 0 1 2 3 x i Scale 2 –6 –4 –2 0 2 6 4 x i 2.8 We first read the data file “running time” in R > require(xlsx) > running<-na.omit(read.xlsx("C:/Data_sets/running_time.xlsx",1, header=TRUE)) Let us print the first observations > head(running) week time 1 1 12.80 2 2 12.20 3 3 12.25 4 4 12.18 5 5 11.53 6 6 12.47 The mean 1-mile running time over 18 weeks is equal to 12.09 minutes: > mean(running$time) [1] 12.08889 2.9 The standard deviation is given by > sd(running$time) [1] 0.3874181 2.10 Let us first create the variable “time_100” and then calculate its mean and standard deviation > running$time_100=100*running$time > mean(running$time_100) [1] 1208.889 > sd(running$time_100) [1] 38.74181 2.11 Let us to construct the stem-and-leaf plot in R using the stem.leaf command from the package “aplpack” > require(aplpack)

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    CHAPTER 2/ DESCRIPTIVE STATISTICS 4 > stem.leaf(running$time_100, unit=1, trim.outliers=FALSE) 1 | 2: represents 12 leaf unit: 1 n: 18 2 115 | 37 3 116 | 7 5 117 | 23 7 118 | 03 8 119 | 2 (1) 120 | 8 9 121 | 8 8 122 | 05 6 123 | 03 4 124 | 7 3 125 | 5 2 126 | 7 127 | 1 128 | 0 Note: one can also use the standard command stem (which does require the “aplpack” package) to get a similar plot > stem(running$time_100, scale = 4) 2.12 The quantiles of the running times are > quantile(running$time) 0% 25% 50% 75% 100% 11.5300 11.7475 12.1300 12.3225 12.8000 An outlying value is identify has any value x such that x  upper quartile+1.5  ( upper quartile-lower quartile)  12.32  1.5  (12.32  11.75)  12.32  0.85  13.17 Since 12.97 minutes is smaller than the largest nonoutlying value (13.17 minutes), this running time recorded in his first week of running in the spring is not an outlying value relative to the distribution of running times recorded the previous year. 2.13 The mean is x x i     24 469 24 19 54 . mg dL 2.14 We have that s 2  ( x i  x i  1 24  ) 2 23  (49  19.54) 2    (12  19.54) 23 2  6495.96 23  282.43 s  282.43  16.81 mg/dL 2.15 We provide two rows for each stem corresponding to leaves 5-9 and 0-4 respectively. We have 11.6 11.8 12.0 12.2 12.4 12.6 12.8 Box plot of running times Time

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    CHAPTER 2/ DESCRIPTIVE STATISTICS 5 Stem-and- leaf plot Cumulative frequency +4 98 24 +4 1 22 +3 65 21 +3 21 19 +2 78 17 +2 13 15 +1 9699 13 +1 332 9 +0 88 6 +0 2 4  0  0 8 3  1 03 2 2.16 We wish to compute the average of the (24/2)th and (24/2 + 1)th largest values  average of the 12th and 13th largest points. We note from the stem-and-leaf plot that the 13th largest point counting from the bottom is the largest value in the upper  1 row  19. The 12th largest point  the next largest value in this row  19. Thus, the median    19 19 2 19 mg dL . 2.17 We first must compute the upper and lower quartiles. Because 24 75 100 18    is an integer, the upper quartile = average of the 18th and 19th largest values    32 31 2 31 5 . . Similarly, because 24 25 100 6    is an integer, the lower quartile  average of the 6th and 7th smallest points    8 12 2 10 . Second, we identify outlying values. An outlying value is identified as any value x such that x upper quartile upper quartile lower quartile)            1 5 31 5 1 5 31 5 10 31 5 32 25 63 75 . ( . . ( . ) . . . or x lower quartile upper quartile lower quartile)             15 10 1 5 315 10 10 32 25 22 25 . ( . ( . ) . . From the stem-and-leaf plot, we note that the range is from  13 to  49. Therefore, there are no outlying values. Thus, the box plot is as follows: Stem-and- leaf plot Cumulative frequency Box plot +4 98 24 | +4 1 22 | +3 65 21 | +3 21 19        +2 78 17 | | +2 13 15 | | +1 9699 13        +1 332 9        +0 88 6 | +0 2 4 |  0 |  0 8 3 |  1 03 2 |

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    CHAPTER 2/ DESCRIPTIVE STATISTICS 6 Comments: The distribution is reasonably symmetric, since the mean    19 54 19 .  mg dL mg dL median. This is also manifested by the percentiles of the distribution since the upper quartile     median 31 5 19 12 5 . .       median lower quartile 19 10 9 . The box plot looks deceptively asymmetric, since 19 is the highest value in the upper  1 row and 10 is the lowest value in the lower  1 row. 2.18 To compute the median cholesterol level, we construct a stem-and-leaf plot of the before-cholesterol measurements as follows. Stem-and- leaf plot Cumulative frequency 25 0 24 24 4 23 23 68 22 22 42 20 21 20 5 18 19 5277 17 18 0 13 17 8 12 16 698871 11 15 981 5 14 5 2 13 7 1 Based on the cumulative frequency column, we see that the median  average of the 12th and 13th largest values    178 180 2 179 mg/dL. Therefore, we look at the change scores among persons with baseline cholesterol  179 mg/dL and < 179 mg/dL, respectively. A stem-and-leaf plot of the change scores in these two groups is given as follows: Baseline  179 mg/dL Baseline < 179 mg/dL Stem-and- leaf plot Stem-and- leaf plot +4 98 +4 +4 +4 1 +3 65 +3 +3 2 +3 1 +2 78 +2 +2 1 +2 3 +1 699 +1 9 +1 +1 332 +0 8 +0 8 +0 +0 2  0  0  0  0 8  1  1 03 Clearly, from the plot, the effect of diet on cholesterol is much greater among individuals who start with relatively high cholesterol levels (  179 mg/dL) versus those who start with relatively low levels (< 179 mg/dL). This is also evidenced by the mean change in cholesterol levels in the two groups, which is 28.2 mg/dL in the  179 mg/dL group and 10.9 mg/dL in the < 179 mg/dL group. We will be discussing the formal statistical methods for comparing mean changes in two groups in our work on two- sample inference in Chapter 8.

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    CHAPTER 2/ DESCRIPTIVE STATISTICS 7 2.19 We first calculate the difference scores between the two positions: Subject number Subject Systolic difference score Diastolic difference score 1 B.R.A.   6   8 2 J.A.B. +2   2 3 F.L.B. +6 +4 4 V.P.B. +8   4 5 M.F.B. +8 +2 6 E.H.B. +12 +4 7 G.C. +10 0 8 M.M.C. 0   2 9 T.J.F.   2   8 10 R.R.F. +4   2 11 C.R.F. +8   2 12 E.W.G. +14 +4 13 T.F.H. +2  14 14 E.J.H. +6   2 15 H.B.H. +26 0 16 R.T.K. +8 +8 17 W.E.L. +10 +4 18 R.L.L. +12 +2 19 H.S.M. +14 +8 20 V.J.M.   8   2 21 R.H.P. +10 +14 22 R.C.R. +14 +4 23 J.A.R. +14 0 24 A.K.R. +4 +4 25 T.H.S. +6 +4 26 O.E.S. +16 +2 27 R.E.S. +28 +16 28 E.C.T. +18   4 29 J.H.T. +14 +4 30 F.P.V. +4   6 31 P.F.W. +12 +6 32 W.J.W. +8   4 Second, we calculate the mean difference scores: x sys mm Hg       6 8 32 282 32 8 8  . x dias mm Hg          8 4 32 30 32 0 9  . The median difference scores are given by the average of the 16th and 17th largest values. Thus, median mm Hg sys    8 8 2 8 median mm Hg dias    0 2 2 1

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    CHAPTER 2/ DESCRIPTIVE STATISTICS 8 2.20 The stem-and-leaf and box plots allowing two rows for each stem are given as follows: Systolic Blood Pressure Stem-and- leaf plot Cumulative frequency Box plot 2 68 32 | 2 | 1 68 30 | 1 20402404442 28        0 68886868 17        0 204244 9         0 2 3 |  0 68 2 | Median upper quartile lower quartile        8 14 14 2 14 4 4 2 4 , , , outlying values: x x            14 15 14 4 29 4 15 14 4 11 . ( ) . ( ) or . Since the range of values is from –8 to +28, there are no outlying values for systolic blood pressure. Diastolic Blood Pressure Stem-and- leaf plot Cumulative frequency Box plot 1 6 32 0 1 4 31 0 0 886 30 | 0 42404042404424 27    +     0 242222244 13         0 886 4 |  1 4 1 0 Median upper quartile lower quartile          1 4 4 2 4 2 2 2 2 , , , outlying values: x x             4 1 5 4 2 13 0 2 1 5 4 2 11 0 . ( ) . . ( ) . or . The values +16, +14 and –14 are outlying values. 2.21 Systolic blood pressure clearly seems to be higher in the supine (recumbent) position than in the standing position. Diastolic blood pressure appears to be comparable in the two positions. The distributions are each reasonably symmetric. 2.22 The upper and lower deciles for postural change in systolic blood pressure (SBP) are 14 and 0. Thus, the normal range for postural change in SBP is 0 14 x   . The upper and lower deciles for postural change in diastolic blood pressure (DBP) are 8 and –6. Thus, the normal range for postural change in DBP is 6 8 x    . 2.23 Id Age FEV Hgt Sex Smoke 301 9 1.708 57 0 0 451 8 1.724 67.5 0 0 ...... 61951 15 2.278 60 0 1 63241 16 4.504 72 1 0 71141 17 5.638 70 1 0 71142 16 4.872 72 1 1 73041 16 4.27 67 1 1 73042 15 3.727 68 1 1 73751 18 2.853 60 0 0

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    CHAPTER 2/ DESCRIPTIVE STATISTICS 9 75852 16 2.795 63 0 1 77151 15 3.211 66.5 0 0 MEAN 9.931193 2.63678 61.14358 0.513761 0.099388 MEDIAN 10 2.5475 61.5 SD 2.953935 0.867059 5.703513 18 15 12 9 6 3 90 80 70 60 50 40 30 20 10 0 Age Frequency Histogram of Age 6 5 4 3 2 1 FEV Boxplot of FEV 75 70 65 60 55 50 45 Hgt Boxplot of Hgt 1 0 350 300 250 200 150 100 50 0 Sex Count Chart of Sex 1 0 600 500 400 300 200 100 0 Smoke Count Chart of Smoke 2.24 Results for Sex = 0 Variable Age Mean StDev Minimum Median Maximum FEV 3 1.0720 * 1.0720 1.0720 1.0720 4 1.316 0.290 0.839 1.404 1.577 5 1.3599 0.2513 0.7910 1.3715 1.7040 6 1.6477 0.2182 1.3380 1.6720 2.1020 7 1.8330 0.3136 1.3700 1.7420 2.5640 8 2.1490 0.4046 1.2920 2.1900 2.9930 9 2.3753 0.4407 1.5910 2.3810 3.2230 10 2.6814 0.4304 1.4580 2.6895 3.4130 11 2.8482 0.4293 2.0810 2.8220 3.7740 12 2.9481 0.3679 2.3470 2.8890 3.8350 13 3.0656 0.4321 2.2160 3.1135 3.8160 14 2.962 0.383 2.236 2.997 3.428 15 2.761 0.415 2.198 2.783 3.330 16 3.058 0.397 2.608 2.942 3.674 17 3.5000 * 3.5000 3.5000 3.5000 18 2.9470 0.1199 2.8530 2.9060 3.0820 19 3.4320 0.1230 3.3450 3.4320 3.5190 Results for Sex = 1 Variable Age Mean StDev Minimum Median Maximum FEV 3 1.4040 * 1.4040 1.4040 1.4040 4 1.196 0.524 0.796 1.004 1.789 5 1.7447 0.2336 1.3590 1.7920 2.1150 6 1.6650 0.2304 1.3380 1.6580 2.2620

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    CHAPTER 2/ DESCRIPTIVE STATISTICS 10 7 1.9117 0.3594 1.1650 1.9050 2.5780 8 2.0756 0.3767 1.4290 2.0690 2.9270 9 2.4822 0.5086 1.5580 2.4570 3.8420 10 2.6965 0.6020 1.6650 2.6080 4.5910 11 3.2304 0.6459 1.6940 3.2060 4.6370 12 3.509 0.871 1.916 3.530 5.224 13 4.011 0.690 2.531 4.045 5.083 14 3.931 0.635 2.276 3.882 4.842 15 4.289 0.644 3.727 4.279 5.793 16 4.193 0.437 3.645 4.270 4.872 17 4.410 1.006 3.082 4.429 5.638 18 4.2367 0.1597 4.0860 4.2200 4.4040 19 5.1020 * 5.1020 5.1020 5.1020 ----------------------------------------------------------------------------------------------------------------------------- Results for Sex = 0 Variable Hgt Mean FEV 46.0 1.0720 46.5 1.1960 48.0 1.110 49.0 1.4193 50.0 1.3378 51.0 1.5800 51.5 1.474 52.0 1.389 52.5 1.577 53.0 1.6887 53.5 1.4150 54.0 1.6408 54.5 1.7483 55.0 1.6313 55.5 2.036 56.0 1.651 56.5 1.7875 57.0 1.9037 57.5 1.9300 58.0 2.1934 58.5 1.9440 59.0 2.1996 59.5 2.517 60.0 2.5659 60.5 2.5563 61.0 2.6981 61.5 2.626 62.0 2.7861 62.5 2.7777 63.0 2.7266 63.5 2.995 64.0 2.9731 64.5 2.864 65.0 3.090 65.4 2.4340 65.5 3.154 66.0 2.984 66.5 3.2843 67.0 3.167 67.5 2.922 68.0 3.214 68.5 3.3300 69.5 3.8350 71.0 2.5380 20 15 10 5 20 15 10 5 6 4 2 70 60 50 6 4 2 70 60 50 Age, 0 FEV Age, 1 Hgt, 0 Hgt, 1 Scatterplot of FEV vs Age, Hgt Panel variable: Sex

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    CHAPTER 2/ DESCRIPTIVE STATISTICS 11 Results for Sex = 1 Variable Hgt Mean FEV 47.0 0.981 48.0 1.270 49.5 1.4250 50.0 1.794 50.5 1.536 51.0 1.683 51.5 1.514 52.0 1.5915 52.5 1.7100 53.0 1.6646 53.5 1.974 54.0 1.7809 54.5 1.8380 55.0 1.8034 55.5 1.8070 56.0 2.025 56.5 1.879 57.0 2.0875 57.5 1.829 58.0 2.0169 58.5 2.131 59.0 2.350 59.5 2.515 60.0 2.279 60.5 2.3253 61.0 2.4699 61.5 2.5410 62.0 2.658 62.5 2.829 63.0 2.877 63.5 2.757 64.0 2.697 64.5 3.100 65.0 2.770 65.5 3.0343 66.0 3.115 66.5 3.353 67.0 3.779 67.5 3.612 68.0 3.878 68.5 3.872 69.0 4.022 69.5 3.743 70.0 4.197 70.5 3.931 71.0 4.310 71.5 4.7200 72.0 4.361 72.5 4.2720 73.0 5.255 73.5 3.6450 74.0 4.654 --------------------------------------------------------------------------------------------------------------------------------- Descriptive Statistics: FEV Results for Sex = 0

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    CHAPTER 2/ DESCRIPTIVE STATISTICS 12 Variable Smoke Mean StDev FEV 0 2.3792 0.6393 1 2.9659 0.4229 Results for Sex = 1 Variable Smoke Mean StDev FEV 0 2.7344 0.9741 1 3.743 0.889 2.25 Looking at the scatterplot of FEV vs. Age, we find that FEV increases with age for both boys and girls, at approximately the same rate. However, the spread (standard deviation) of FEV values appears to be higher in male group than in the female group. 2.26 Variable Mean StDev Median Sat. Fat - DR 14.557 7.536 12.000 Sat. Fat - FFQ 7.898 9.695 3.159 Tot. Fat - DR 64.238 9.894 63.500 Tot. Fat - FFQ 15.21 27.00 1.00 Alcohol - DR 2.470 6.314 0.000 Alcohol - FFQ 8.951 12.255 4.550 Calories - DR 1619.9 323.4 1606.0 Calories - FFQ 1371.7 482.1 1297.6 Alcohol - FFQ Alcohol - DR Tot. Fat - FFQ Tot. Fat - DR Sat. Fat - FFQ Sat. Fat - DR 140 120 100 80 60 40 20 0 Data Boxplot of Sat. Fat, Tot. Fat, and Alcohol 2.27 Sex Smoke 1 0 1 0 1 0 6 5 4 3 2 1 FEV 0 1 Sex Boxplot of FEV Calories - FFQ Calories - DR 3000 2500 2000 1500 1000 500 Data Boxplot of Calories

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    CHAPTER 2/ DESCRIPTIVE STATISTICS 13 60 45 30 15 0 50 40 30 20 10 150 100 50 0 120 100 80 60 40 60 45 30 15 0 48 36 24 12 0 3000 2000 1000 2500 2000 1500 1000 Sat. Fat - DR*Sat. Fat - FFQ Tot. Fat - DR*Tot. Fat - FFQ Alcohol - DR*Alcohol - FFQ Calories - DR*Calories - FFQ Scatterplot of DR vs. FFQ values If FFQ were a perfect substitute for DR, the points would line up in a straight line. If the two were unrelated, then we would expect to see a random pattern in each panel. The scatterplots shown above seem to suggest that the DR and FFQ values are not highly related. 2.28 The 5x5 tables below show the number of people classified into a particular combination of quintile categories. For each table, the rows represent the quintiles of the DR, and the columns represent quintiles of the FFQ. Overall, we get the same impression that there is weak concordance between the two measures. However, we do notice that the agreement is greatest for the two measures with regards to alcohol consumption. Also, we note the relatively high level of agreement at the extremes of each nutrient; for example, the (1,1) and (5,5) cells generally contain the highest values. Tabulated statistics: SFDQuin, SFFQuin Rows: SFDQuin Columns: SFFQuin 1 2 3 4 5 All 1 15 8 9 2 1 35 2 10 6 6 8 5 35 3 4 7 8 9 6 34 4 6 10 6 9 4 35 5 0 3 6 7 18 34 All 35 34 35 35 34 173 Cell Contents: Count Tabulated statistics: TFDQuin, TFFQuin Rows: TFDQuin Columns: TFFQuin 1 2 3 4 5 All 1 13 9 8 5 1 36 2 9 5 7 10 3 34 3 4 10 8 6 6 34 4 8 6 3 9 9 35 5 1 5 8 5 15 34

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    CHAPTER 2/ DESCRIPTIVE STATISTICS 14 All 35 35 34 35 34 173 Cell Contents: Count Tabulated statistics: AlcDQuin, AlcFQuin Rows: AlcDQuin Columns: AlcFQuin 1 2 3 4 5 All 1 28 5 2 0 0 35 2 6 23 6 0 0 35 3 0 9 14 10 1 34 4 0 1 10 16 8 35 5 0 0 0 8 26 34 All 34 38 32 34 35 173 Cell Contents: Count Tabulated statistics: CalDQuin, CalFQuin Rows: CalDQuin Columns: CalFQuin 1 2 3 4 5 All 1 10 11 8 4 2 35 2 11 4 9 7 4 35 3 5 9 6 8 6 34 4 4 8 7 6 10 35 5 5 3 4 10 12 34 All 35 35 34 35 34 173 2.29 Descriptive Statistics: Total Fat Density DR, Total Fat Density FFQ Variable Mean StDev Median Total Fat Density DR 38.066 4.205 38.646 Total Fat Density FFQ 36.855 6.729 36.366 60 50 40 30 20 10 0 50 40 30 20 10 0 Total Fat Density FFQ Total Fat Density DR 0 0 Scatterplot of Total Fat Density DR vs Total Fat Density FFQ

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