Assignment
Afshin Motavali
June, 2022
Question
The model of interest is a linear-regression model with two predictors, haemoglobin (g/dL) and serum cholesterol (mg/dL) that we know affect our outcome of interest – body mass index (BMI; kg/m2). We are interested to know how many individuals we are required to sample, in order to detect a change of 0.5 in BMI for individuals who received a particular treatment (e.g., an exercise program) compared to those who did not (i.e., “treatment” is the third predictor in the model). Equation below, presents the model of interest:
BMI = β0 + β1Hgb + β2Chol + β3*Treatment + ε
Where: + β0 = 18.06, β1= 0.297, β2=0.014; + Haemoglobin values (Hgb; g/dL) are normally distributed with a mean of 14.4 g/dL and standard deviation of 1.5 g/dL (i.e. N(14.4,1.5) distributed); + Serum Cholesterol values (Chol; mg/dL) are N(215,50) distributed; and + Treatment is an indicator variable, where 1 represents those that received the treatment, and 0 represents those that did not; and + The error term, ε ~ N(0,1).
- For this question,
- Write code that:
- randomly samples 100 individual’s haemoglobin and serum cholesterol from the above distributions§,
- randomly allocates exactly half of the subjects to the Treatment group, and
- calculates their BMI according to the above equation, assuming that β3=0.5.
Store the simulated haemoglobin, serum cholesterol, treatment, and BMI values in a data frame/dataset. §Note: To generate random normal values, you will need to use rnorm(). To randomly allocate individuals to each treatment group, you will need to use sample().
- Using your code from Part C Q1a, write a function which takes as input the value for the sample size, n, and returns an object/data frame containing n simulated individuals with their haemoglobin, serum cholesterol, treatment, and BMI values.
Using your code from parts a and b, modify the code and include a for loop that:
Simulates the model 1000 times; and
fits a linear regression model* of “BMI ~ Hgb + Chol + Treatment” for each simulated data set, and counts the number of times that the p-value associated with Treatment is <0.05; and
cycles through sample sizes between 40 and 200 in step sizes of 20 and performs (a) and (b) and stores/returns the proportion of simulated experiments that had a p- value for treatment <0.05. This is an estimate of the Power (i.e., 1 – Type II error rate). *Notes: To fit the linear regression model in R, you can use:
lm(BMI ~ Hgb + Chol + Trt, data=df)
Make sure that you set the random number seed so that the results are reproducible.
- Create a figure showing the sample size on the x-axis, and the estimated Power on the y- axis. Include the following items in your figure:
A horizontal line at 80% Power.
Use your favourite colour (ensure that this is not the default colour) to highlight the point that corresponds to the smallest sample size such that “we have at least 80% power to detect a difference of 0.5 in treatment between the two groups at the 5% significance level”. You will need to automate this, and not manually select this value (Hint: you want to find the minimum sample size that has an estimated Power ≥80%).
Change the y-axis to have labels from 20% to 100% power in steps of 10%. Change the x-axis to go from a sample size of 40 to 200 in steps of 40. Make sure the axes have appropriate labels.
Solution
#--------------------------
# 1
#-------------------------
n = 100
b0 = 18.06
b1 = 0.297
b2 = 0.014
## a
# 1 - a - 1
set.seed(2022)
Hgb <- rnorm(n, mean=14.4, sd=1.5)
Chol <- rnorm(n, mean=215, sd=50)
epsilon <- rnorm(n, mean=0, sd=1)
# 1 - a - 2
Treatment <- sample(c(0,1), size=100, prob=c(0.5,0.5), replace = T)
# 1 - a - 3
b3 = 0.5
BMI = b0 + b1*Hgb + b2*Chol + b3*Treatment + epsilon
# store the simulated data
df <- data.frame(bmi=BMI, hgb=Hgb, chol=Chol, treatment=Treatment)
## b
simulate_data <- function(n){
b0 = 18.06
b1 = 0.297
b2 = 0.014
b3 = 0.5
set.seed(2022)
Hgb <- rnorm(n, mean=14.4, sd=1.5)
Chol <- rnorm(n, mean=215, sd=50)
epsilon <- rnorm(n, mean=0, sd=1)
Treatment <- sample(c(0,1), size=n, prob=c(0.5,0.5), replace = T)
BMI = b0 + b1*Hgb + b2*Chol + b3*Treatment + epsilon
df <- data.frame(bmi=BMI, hgb=Hgb, chol=Chol, treatment=Treatment)
return(df)
}
simulate_data(n=500)## bmi hgb chol treatment
## 1 25.12694 15.750213 212.35791 1
## 2 26.54286 12.639981 240.62834 1
## 3 25.10047 13.053772 162.31710 1
## 4 24.41227 12.233248 189.44775 0
## 5 27.05235 13.903480 274.56931 1
## 6 24.14776 10.049057 149.53850 0
## 7 27.55383 12.811116 199.35078 1
## 8 22.53127 14.816932 202.23302 0
## 9 25.17493 15.524229 294.90889 0
## 10 27.99024 14.762374 242.85593 0
## 11 25.13902 15.909279 163.02917 1
## 12 28.51179 14.122281 191.76590 1
## 13 26.87822 12.927260 239.68633 0
## 14 24.80586 14.539362 212.21367 0
## 15 26.46931 14.320823 257.51184 0
## 16 24.17313 14.279508 97.01461 1
## 17 26.22904 13.418844 240.67956 0
## 18 24.82255 12.973975 188.80500 0
## 19 24.97944 15.929343 212.93683 0
## 20 24.12676 15.688570 209.04146 0
## 21 25.82677 14.946691 165.30374 1
## 22 26.26465 14.975477 227.09393 0
## 23 24.86658 16.070109 205.48847 0
## 24 27.81664 16.217265 298.12505 1
## 25 25.21722 13.877512 211.16308 1
## 26 24.70479 13.110670 180.60880 1
## 27 25.89369 15.375041 268.12742 1
## 28 26.80592 14.892089 208.12400 1
## 29 26.87685 13.623080 169.05866 1
## 30 25.67661 14.041527 210.02768 1
## 31 27.87679 14.576668 294.34492 1
## 32 27.14023 15.647278 191.16643 1
## 33 25.47028 12.061622 273.24845 1
## 34 24.57900 14.069221 207.79610 0
## 35 24.88387 13.174208 179.10065 0
## 36 27.52208 16.015016 228.96288 0
## 37 26.75532 16.019496 202.78367 0
## 38 24.25118 14.613192 197.02448 0
## 39 27.82632 14.635469 276.43649 1
## 40 24.87713 14.146920 176.27698 1
## 41 27.28721 13.996443 238.59048 1
## 42 25.30310 15.611653 226.23785 1
## 43 21.87393 12.712924 133.87104 0
## 44 23.81523 12.253818 197.72247 0
## 45 24.08607 14.490535 241.88040 0
## 46 25.75813 13.210526 185.04041 0
## 47 26.30597 14.910414 230.43212 1
## 48 26.17075 14.010797 222.97754 0
## 49 25.45835 12.442727 274.09508 0
## 50 25.08306 14.952260 158.08106 0
## 51 26.62039 16.939785 293.79321 0
## 52 24.80317 15.893756 135.19058 0
## 53 27.16724 14.680128 262.28356 0
## 54 28.07891 16.257506 319.45468 1
## 55 25.44699 14.864060 247.68251 0
## 56 26.18495 15.353577 249.89418 1
## 57 26.40906 14.434775 168.73013 1
## 58 25.94033 16.166795 221.39347 0
## 59 24.40747 13.719680 195.81778 0
## 60 26.45487 15.023891 190.75270 1
## 61 23.80299 12.392336 148.73038 0
## 62 23.22571 12.462038 192.70874 1
## 63 24.70963 13.936389 195.58639 0
## 64 24.97891 14.634768 172.28817 1
## 65 25.20963 13.149125 202.81510 1
## 66 27.08536 14.363176 279.21920 0
## 67 25.81796 12.693973 263.96319 1
## 68 24.77774 16.008081 192.76090 1
## 69 27.57601 17.871748 213.28831 1
## 70 25.60357 15.034460 181.20932 1
## 71 27.71422 14.194591 235.04531 1
## 72 26.11415 16.392595 168.02782 1
## 73 26.28365 15.054803 181.83112 1
## 74 25.01364 14.499643 227.86948 0
## 75 24.71555 16.356990 238.04497 0
## 76 25.46675 14.086111 182.02682 1
## 77 28.52671 15.927380 279.70852 1
## 78 25.76202 16.449052 185.44585 1
## 79 25.20227 16.615044 150.71046 0
## 80 27.20999 15.730960 270.11261 0
## 81 26.14446 12.876137 191.41845 1
## 82 26.21822 17.206304 243.18166 1
## 83 26.02068 16.014298 182.25876 1
## 84 24.73643 12.788389 179.66865 0
## 85 23.74992 11.106636 169.02565 1
## 86 27.69858 15.201817 234.53593 1
## 87 28.29847 16.415600 282.88463 1
## 88 27.93868 16.477505 201.86881 1
## 89 25.87495 18.520390 282.85236 1
## 90 23.47309 14.331083 222.44028 0
## 91 24.98898 15.514628 269.00950 0
## 92 26.78012 14.790637 256.20213 1
## 93 24.52912 15.042423 279.86864 1
## 94 25.54529 13.847612 274.89570 0
## 95 26.02619 18.731135 178.54533 1
## 96 26.01967 13.489391 205.05538 1
## 97 24.01067 11.581202 202.34870 1
## 98 26.94961 15.477535 196.98109 0
## 99 26.03338 14.777138 173.98172 0
## 100 27.11405 15.100534 278.37514 1
## 101 24.85311 14.073167 233.78677 0
## 102 26.17482 12.573528 207.62336 1
## 103 25.41286 13.268882 212.41006 1
## 104 26.32711 14.686845 187.96290 1
## 105 25.76923 15.970924 167.99604 1
## 106 24.05369 16.034743 192.69394 1
## 107 24.78739 15.443212 174.85641 1
## 108 24.56023 13.613447 204.50803 0
## 109 26.72097 14.796024 213.54635 1
## 110 26.34472 16.261801 188.45974 0
## 111 24.24901 15.364946 253.03779 0
## 112 25.13098 12.904856 167.46343 1
## 113 24.78394 11.396552 255.67459 1
## 114 26.52156 13.686241 164.11634 0
## 115 26.50510 14.466270 248.37817 0
## 116 26.58018 12.328297 287.06294 1
## 117 25.92608 14.771279 217.66557 0
## 118 23.91203 14.736284 203.74326 1
## 119 27.30881 15.692218 243.16856 1
## 120 25.32708 13.009857 245.63389 0
## 121 25.93535 13.919335 259.13232 0
## 122 25.69387 15.409537 203.02439 1
## 123 24.20011 12.501555 235.06158 0
## 124 24.53324 13.622219 176.97483 1
## 125 25.11006 16.083741 278.69155 0
## 126 24.76475 13.926034 192.74710 1
## 127 25.86012 11.762551 204.05257 1
## 128 25.64865 16.255880 252.98314 0
## 129 24.48993 11.904071 140.25345 1
## 130 25.60849 17.183602 243.88590 1
## 131 26.86094 15.986996 213.92534 0
## 132 26.82818 14.455903 249.83580 1
## 133 22.41061 14.406908 160.10030 0
## 134 24.97480 11.484249 221.29088 0
## 135 25.02036 12.045565 228.43541 0
## 136 24.34246 13.990054 111.56698 0
## 137 22.56199 15.103281 128.55934 0
## 138 25.55650 15.948808 199.48294 0
## 139 26.76196 12.272406 238.29941 1
## 140 25.53661 14.268184 153.63230 1
## 141 28.25597 16.767946 253.44926 1
## 142 25.32184 14.766937 203.52560 1
## 143 27.60008 16.645640 210.19184 0
## 144 24.89702 15.773860 228.02142 1
## 145 26.20008 14.881262 222.41113 1
## 146 25.72992 15.244086 236.81337 1
## 147 26.76597 15.208943 204.74280 0
## 148 24.14640 12.789893 194.66957 0
## 149 26.73724 11.575259 226.18036 1
## 150 27.28330 15.767085 239.29117 0
## 151 24.67972 14.474395 202.96817 0
## 152 22.53689 12.546518 143.85339 1
## 153 25.11921 12.504372 101.85439 0
## 154 26.78078 12.685703 249.02604 1
## 155 26.11958 15.381618 250.49868 1
## 156 25.44986 13.384771 255.67208 1
## 157 25.06219 12.789134 252.41474 1
## 158 25.38574 12.936513 242.07715 0
## 159 23.78043 12.710588 218.35665 0
## 160 23.77847 11.446031 240.71192 0
## 161 25.59067 13.527017 242.93569 0
## 162 23.22829 11.756239 160.19386 0
## 163 25.00157 13.608261 192.50645 1
## 164 25.79880 12.338509 235.90843 1
## 165 27.61478 11.499431 316.46745 1
## 166 26.42999 15.869675 203.98146 1
## 167 25.95979 12.156466 305.15574 0
## 168 25.93172 16.951214 188.51392 0
## 169 22.64081 12.298461 145.92915 1
## 170 24.06584 13.673305 200.29956 1
## 171 26.39119 12.575732 247.20315 1
## 172 26.56508 14.557156 181.06405 1
## 173 23.87161 14.036194 196.13570 0
## 174 25.51213 12.276328 251.49210 0
## 175 23.51617 12.825810 190.31435 0
## 176 25.99334 14.803585 283.70783 0
## 177 25.83120 13.681926 200.60504 1
## 178 24.31176 13.950827 189.35118 0
## 179 24.67250 15.957223 181.95621 1
## 180 26.85467 15.943388 198.07853 0
## 181 24.86072 11.442487 262.13010 0
## 182 25.49572 12.475005 168.03509 1
## 183 23.75268 14.472872 178.47896 0
## 184 25.63985 16.215947 234.04577 0
## 185 23.18311 13.672994 124.39698 0
## 186 23.78918 13.210036 213.63432 1
## 187 26.13886 14.658622 201.29242 1
## 188 27.63023 16.057820 240.17966 1
## 189 25.73085 15.511313 194.87021 0
## 190 24.99189 14.037830 108.92948 1
## 191 27.82760 15.322436 177.56409 1
## 192 25.92141 16.473820 162.74332 1
## 193 25.45049 13.525196 359.60610 1
## 194 24.14701 14.103105 75.67260 0
## 195 24.83723 13.495156 231.21584 1
## 196 25.19157 12.767027 224.98404 0
## 197 26.11193 14.675873 260.64648 1
## 198 27.22023 16.360704 173.19966 0
## 199 26.19999 14.147575 219.01978 1
## 200 24.19164 14.915761 167.57821 1
## 201 25.74684 14.964712 175.17961 1
## 202 26.23002 14.108191 156.71149 1
## 203 23.96584 11.226996 166.42427 1
## 204 24.34919 13.030399 148.33683 1
## 205 24.14415 14.924564 155.52941 0
## 206 24.35622 15.008366 212.04533 1
## 207 25.93899 15.157989 156.01390 0
## 208 24.10531 11.712199 252.42904 1
## 209 25.92668 13.407767 234.83792 1
## 210 26.85944 13.841591 236.83735 1
## 211 26.30344 14.637992 261.75685 0
## 212 24.74404 13.382381 151.77747 0
## 213 25.58466 14.523734 204.01978 0
## 214 25.66031 16.265610 239.29369 0
## 215 26.88507 15.720743 270.35970 0
## 216 24.30535 15.215938 174.84747 1
## 217 25.50294 13.213642 193.86321 1
## 218 25.89383 15.040717 170.71503 0
## 219 27.72298 15.826671 261.40167 0
## 220 24.21762 15.630176 248.47175 0
## 221 26.31956 11.683581 303.38230 1
## 222 25.47482 13.980130 244.92406 0
## 223 28.31978 15.406602 288.42855 0
## 224 27.27134 16.795459 296.73760 1
## 225 24.43755 13.311402 185.41862 0
## 226 25.99392 15.840863 82.93915 0
## 227 26.66466 15.834659 189.92007 0
## 228 26.60558 16.298772 256.60362 1
## 229 26.03034 14.044566 235.37528 1
## 230 26.45130 12.119829 255.28710 0
## 231 24.89654 14.909330 115.74684 1
## 232 25.25628 16.493961 196.74873 1
## 233 25.54951 12.620907 289.91509 0
## 234 25.04806 13.793293 180.83857 1
## 235 26.54082 13.969341 163.07598 0
## 236 24.96268 15.813879 228.03589 0
## 237 24.42648 12.817892 173.63636 0
## 238 25.43365 13.414084 248.52279 1
## 239 25.96657 13.778567 280.72052 1
## 240 25.96564 15.080950 229.98020 1
## 241 24.62435 13.569069 144.37158 1
## 242 25.53551 14.930803 218.92990 0
## 243 25.12232 14.622429 130.37516 1
## 244 25.18927 14.964768 167.11732 1
## 245 25.56304 16.660481 212.31213 1
## 246 26.23981 13.009058 162.94262 0
## 247 22.76051 13.519288 117.32131 0
## 248 29.86088 18.088389 352.41460 0
## 249 25.10015 15.410613 154.81660 0
## 250 24.90765 14.901637 182.19263 0
## 251 26.00313 16.813563 183.84139 0
## 252 25.92759 12.726901 185.52510 0
## 253 25.70381 14.415195 252.45534 0
## 254 22.15387 12.821463 176.14678 1
## 255 25.93758 15.050885 244.48781 1
## 256 26.99361 15.387134 351.85845 1
## 257 25.93393 12.239890 244.04435 1
## 258 26.56293 15.110042 196.22031 0
## 259 25.68157 13.191222 311.28527 0
## 260 26.51588 13.221801 258.49011 1
## 261 25.41013 12.696974 246.83677 0
## 262 26.25473 13.032862 268.48178 0
## 263 22.85157 14.534949 233.93294 1
## 264 26.54292 14.045858 219.55317 1
## 265 27.40052 16.841369 263.27061 0
## 266 26.11784 15.157801 294.65391 1
## 267 26.35655 14.456156 267.29144 0
## 268 26.17479 14.947748 270.35960 1
## 269 27.05987 12.308099 277.23651 1
## 270 26.94352 15.352702 229.88337 1
## 271 25.22548 13.683371 239.21822 0
## 272 25.68475 13.170039 286.42646 1
## 273 24.67040 16.278576 202.65816 0
## 274 24.01600 12.422475 196.99885 0
## 275 25.40775 15.216392 257.32268 0
## 276 26.82978 14.119655 284.70109 1
## 277 24.55246 15.323945 228.97534 0
## 278 26.81283 15.191981 265.82160 1
## 279 27.44180 14.751683 199.14576 1
## 280 25.39444 14.231553 186.27238 0
## 281 27.01123 12.425555 211.48454 1
## 282 26.57081 13.894849 253.68482 1
## 283 26.33283 15.117615 174.59683 1
## 284 22.12643 12.759133 165.04209 0
## 285 26.51809 16.292108 242.15247 1
## 286 25.78063 14.454023 254.42717 1
## 287 26.90760 17.221601 253.82281 1
## 288 23.97897 13.100317 214.25571 1
## 289 25.93289 12.907728 215.27375 0
## 290 24.54191 12.124150 261.70187 1
## 291 26.83797 14.339734 282.75082 0
## 292 24.73686 11.720193 210.56385 0
## 293 26.04323 15.834369 198.09425 0
## 294 25.15737 13.979256 266.46904 0
## 295 24.94434 13.783219 181.19847 1
## 296 26.04133 13.704854 268.85551 0
## 297 25.24225 10.717594 203.46801 1
## 298 25.38229 14.869329 232.29966 0
## 299 24.56544 13.212993 198.78874 1
## 300 25.66030 14.738198 235.41881 0
## 301 24.36059 11.724288 223.91623 0
## 302 24.03015 12.345257 173.94383 1
## 303 25.13186 14.990965 155.68990 1
## 304 25.93316 14.800833 213.43632 0
## 305 27.05718 14.530854 217.37239 0
## 306 25.82012 16.727797 208.47023 1
## 307 26.34381 15.593293 194.70379 1
## 308 25.26551 14.912010 121.66709 1
## 309 24.83422 12.735285 223.47163 1
## 310 25.91513 12.836457 266.68391 0
## 311 26.74743 14.116508 220.41772 0
## 312 26.58914 16.329046 259.71003 1
## 313 23.08084 12.253216 149.89079 1
## 314 26.03574 13.818809 218.74804 0
## 315 27.04156 13.646096 289.89966 1
## 316 25.47795 14.501326 281.85143 0
## 317 24.73271 15.159820 182.29582 0
## 318 24.79988 12.477576 195.10925 0
## 319 24.96179 12.816969 150.47421 0
## 320 24.13306 13.676732 206.40085 1
## 321 26.07924 15.492798 181.25564 0
## 322 25.04205 14.072429 256.78141 0
## 323 25.43241 16.363370 192.19731 1
## 324 27.35508 14.023948 200.14659 1
## 325 23.72201 13.581518 193.21460 0
## 326 26.73142 16.246361 246.61698 0
## 327 24.13992 13.593257 179.20043 1
## 328 25.97739 15.293733 188.88778 0
## 329 24.46091 15.077944 165.38873 1
## 330 25.84589 13.403694 210.09449 0
## 331 24.73122 11.058352 182.60410 0
## 332 26.42164 14.398907 310.91963 0
## 333 25.24254 16.438288 180.17038 0
## 334 25.21008 12.408661 247.64455 0
## 335 26.09147 14.774645 235.98040 1
## 336 24.50504 12.659796 203.00214 0
## 337 28.34227 16.281879 302.12509 0
## 338 28.93940 13.983980 282.35611 1
## 339 26.18734 14.059229 196.93202 0
## 340 24.76878 13.717662 224.63339 1
## 341 26.55980 14.230946 271.35946 1
## 342 23.83952 14.348799 167.59591 0
## 343 23.93490 13.802864 169.04428 0
## 344 23.87446 13.898536 253.85102 0
## 345 25.40340 15.238771 191.08649 0
## 346 23.79106 14.035567 268.48205 0
## 347 24.09937 11.864872 183.67366 1
## 348 26.10779 12.457353 208.05381 1
## 349 26.38500 15.945596 148.12490 1
## 350 27.57700 13.239473 289.01373 1
## 351 25.86783 14.996200 203.16640 1
## 352 23.77355 13.301877 163.12826 1
## 353 25.70310 15.234876 178.84257 0
## 354 26.59248 13.609968 233.19229 1
## 355 25.52574 12.884740 197.16438 1
## 356 27.97448 13.944374 323.73028 1
## 357 25.41802 12.910867 259.35862 1
## 358 26.24686 15.259501 198.72064 1
## 359 27.40985 15.243038 192.28047 1
## 360 26.94480 13.526091 295.81484 0
## 361 24.77131 15.930568 216.86108 0
## 362 25.07896 13.916636 221.78901 1
## 363 24.84648 14.341369 125.29256 1
## 364 25.23240 14.746510 162.02674 0
## 365 25.82350 16.422657 164.15514 1
## 366 26.11146 16.389157 132.70363 1
## 367 25.55314 14.327853 234.51463 1
## 368 25.36341 13.504180 238.49637 0
## 369 24.82967 15.311605 157.96821 0
## 370 26.34754 15.630227 285.27916 0
## 371 25.88939 14.088970 250.92162 0
## 372 23.02664 12.470567 199.68270 0
## 373 24.24550 15.008110 211.80824 0
## 374 26.01238 15.263396 222.97139 1
## 375 24.48186 13.410294 194.24229 1
## 376 22.60694 12.831736 171.66831 0
## 377 24.65339 14.900188 192.18583 1
## 378 24.94628 13.304079 155.69899 0
## 379 23.94492 14.256087 186.69829 1
## 380 26.65404 16.918978 195.72015 0
## 381 24.98886 13.970706 172.71537 0
## 382 25.06223 15.846472 245.46485 1
## 383 23.53906 13.317658 199.39306 0
## 384 26.24936 14.746729 190.26525 0
## 385 25.63469 17.118225 228.11962 1
## 386 25.76029 13.485233 229.29481 1
## 387 24.65846 12.983904 156.81576 0
## 388 23.87136 15.221076 136.21324 0
## 389 25.80994 14.891737 270.76247 0
## 390 26.69780 13.809687 266.91755 0
## 391 23.80069 12.877204 206.70272 0
## 392 28.25263 13.460060 266.08603 1
## 393 25.34532 15.573503 150.37418 1
## 394 25.73986 18.440668 295.85160 0
## 395 26.55562 17.196329 197.13139 0
## 396 24.48463 16.167577 188.85519 0
## 397 23.40254 14.140809 196.12791 1
## 398 24.59377 11.435498 160.44293 1
## 399 25.53650 14.038478 205.69181 1
## 400 25.99610 12.832514 252.98641 1
## 401 28.27090 15.638011 320.02304 0
## 402 24.97730 11.173927 146.66335 0
## 403 26.44080 14.813621 206.27442 1
## 404 27.01760 16.000808 286.39534 1
## 405 27.13315 16.979809 257.99337 1
## 406 25.21919 14.849947 185.84047 0
## 407 25.68120 16.049116 176.91991 1
## 408 26.53463 15.171927 172.89935 1
## 409 23.17480 15.134177 96.47925 0
## 410 26.71146 12.886868 241.54404 1
## 411 25.21165 15.290789 196.45134 1
## 412 23.42672 14.064740 199.61672 1
## 413 27.03965 12.502802 255.39112 1
## 414 24.93401 14.117664 206.62849 0
## 415 24.08589 14.717988 202.11920 0
## 416 24.88508 13.022936 222.46028 0
## 417 27.14197 12.990597 323.81994 0
## 418 23.72648 13.907153 116.20418 1
## 419 29.63747 16.413280 323.04280 1
## 420 22.88077 14.369231 149.28867 0
## 421 26.00685 13.980665 297.60635 0
## 422 25.15046 13.403918 241.87404 0
## 423 27.75636 17.486841 212.09493 1
## 424 25.61391 12.871487 201.13537 0
## 425 26.42952 16.192530 256.04603 0
## 426 28.69186 14.847315 353.35551 0
## 427 25.73878 14.632544 192.32414 0
## 428 26.39262 14.788566 210.29290 1
## 429 24.37401 15.216084 158.24953 0
## 430 24.76091 12.813423 244.69507 1
## 431 26.12852 15.809807 181.29375 0
## 432 28.46161 13.974885 255.61380 0
## 433 23.81119 13.159490 135.39723 0
## 434 25.71240 17.343734 168.52761 1
## 435 23.04653 13.056301 127.15398 1
## 436 25.27853 18.073208 211.94218 0
## 437 24.14865 13.599923 183.94507 0
## 438 26.99562 14.533471 273.09744 1
## 439 28.04716 12.263789 360.78317 1
## 440 27.99184 14.075046 276.29293 1
## 441 29.00691 13.958972 308.69204 1
## 442 25.75267 13.793811 228.62744 1
## 443 26.65006 13.670639 226.30445 0
## 444 24.48259 12.297738 264.48682 0
## 445 25.19378 14.941285 140.79729 1
## 446 30.02507 17.239820 276.70594 1
## 447 26.30879 14.343898 224.16676 1
## 448 24.15242 13.124177 270.61060 0
## 449 26.06933 15.109535 198.78020 0
## 450 25.87033 14.647418 218.49527 1
## 451 25.36814 12.338186 219.91483 0
## 452 24.82058 17.593841 296.81038 0
## 453 25.74990 13.716944 267.65775 0
## 454 23.53976 14.654663 144.62806 0
## 455 24.20400 11.509538 243.21404 1
## 456 23.87439 16.416819 186.36149 0
## 457 24.17707 13.646999 142.12470 0
## 458 25.93571 14.539506 254.12972 0
## 459 23.42563 9.187151 257.31706 0
## 460 28.37411 12.871247 293.44313 1
## 461 26.34841 12.393671 258.51469 1
## 462 25.12550 13.949890 138.15223 1
## 463 27.34674 15.245191 278.51344 0
## 464 23.71500 9.787497 257.71266 1
## 465 26.95586 13.707716 276.02723 0
## 466 25.10601 17.664682 155.80390 0
## 467 25.02304 14.953277 137.11964 1
## 468 23.57642 15.937047 211.19923 1
## 469 24.67981 15.377539 272.68266 0
## 470 27.25027 15.198491 272.66829 0
## 471 25.40680 15.323189 165.65537 1
## 472 27.54866 14.102680 207.60452 1
## 473 26.09885 14.269052 242.74766 1
## 474 26.38810 15.349087 183.71818 1
## 475 26.18298 14.434507 219.87283 1
## 476 25.04984 13.585372 211.25322 0
## 477 27.58656 13.327393 269.49535 0
## 478 22.32275 11.387459 102.50166 0
## 479 25.53536 14.323856 194.58279 0
## 480 24.88320 13.791699 256.28019 1
## 481 26.34697 14.945574 206.74715 0
## 482 27.86171 20.047787 296.71898 0
## 483 25.78034 14.376338 254.95875 0
## 484 26.83404 16.343233 236.92803 0
## 485 26.43990 13.483230 277.53363 1
## 486 25.88610 14.249398 222.81560 0
## 487 26.57274 12.448554 282.73317 1
## 488 25.66364 17.499347 231.79560 1
## 489 23.73441 13.103178 157.65195 1
## 490 24.84372 13.625779 143.36230 1
## 491 26.80982 15.642331 279.58677 1
## 492 24.34680 13.797708 207.50747 0
## 493 27.44017 13.710044 302.07536 0
## 494 27.28606 16.708241 284.82170 1
## 495 26.53782 14.796240 255.09687 1
## 496 24.88231 13.895077 149.66834 0
## 497 25.61486 13.568672 138.87396 1
## 498 25.80763 15.764450 236.49060 1
## 499 26.70328 13.156287 224.69851 1
## 500 25.61063 16.477538 260.91417 0# b
count_tr <- 0
for (i in 1:1000) {
set.seed(2022)
df <- simulate_data(n=100)
fit <- lm(bmi~hgb+chol+treatment, data=df)
s <- summary(fit)
pvalue <- s$coefficients[4,4]
if(pvalue < 0.05){
count_tr <- count_tr + 1
}
}
count_tr## [1] 0## c
sample_sizes <- seq(40,200,by=20)
proportions <- c()
for (s in 1:length(sample_sizes)) {
nn <- sample_sizes[s]
count_tr <- 0
for (i in 1:1000) {
set.seed(2022)
df <- simulate_data(n=nn)
fit <- lm(bmi~hgb+chol+treatment, data=df)
mysummary <- summary(fit)
pvalue <- mysummary$coefficients[4,4]
if(pvalue < 0.05){
count_tr <- count_tr + 1
}
}
proportions[s] <- count_tr/1000
}
proportions## [1] 0 0 1 0 1 1 1 1 1mean(proportions) # mean of power## [1] 0.6666667#--------------------------
# 2
#-------------------------
# 2 - a
plot(sample_sizes, proportions, col='blue', lwd=5)
abline(h=0.80, col='red', lty=3, lwd=3)
# 2- b
# minimum sample size that has power > 0.80
sample_sizes[which(proportions >= 0.80)[1]]## [1] 80# 2 - c
xa = seq(min(sample_sizes), max(sample_sizes), by=40)
ya = seq(min(proportions), max(proportions), length=9)
plot(sample_sizes, proportions, col='blue', lwd=5, xaxt='n', yaxt='n', ylab='Power', xlab='Sample Sizes')
abline(h=0.80, col='red', lty=3, lwd=3)
axis(side=1, at=xa, labels = seq(40,200,by=40))
axis(side=2, at=ya, labels = paste0(seq(0.2,1,by=0.1)*100,'%'))
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