Researchers in toxicology study were interested in evaluating the effects of air

Question

Researchers in toxicology study were interested in evaluating the effects of air pollution on vasoconstriction of small pulmonary arteries in rats. Also, they wanted to know if these health effects differed according to whether the rats had preexisting pulmonary disease. Thus, chronic bronchitis was induced in some of the rats by exposing them to SO2 for 6 weeks prior to pollution exposure. All rats were randomized to one of four groups:

filtered air, not exposed to SO2

concentrated air particles (CAPs), not exposed to SO2

filtered air, exposed to SO2

concentrated air particles (CAPs), exposed to SO2

and the amount of pulmonary inflammation, as measured by neutrophil numerical density (Nn) in each animal was measured after three successive days of air pollution exposure. A large value of Nn denotes pulmonary inflammation. The data below has variables CAPs, SO2 and Nn are named as caps,so2, and nn respectively.

Part 1: Consider the main effect model and the interaction model (which contains both main effects and interactions) that simultaneously assess two categorical factors (SO2 and CAPs). Write down these two regression models, making sure to explicitly define all independent variables. Write out the interpretation of each regression coefficient specified in each of these two models.

Part 2: Consider a test of whether there is a difference in the health effects of air pollution inhalation for healthy animals and that for chronic bronchitic animals (i.e. those who received SO2) under the interaction model. What is the null hypothesis corresponding to this test, in terms of the regression coefficients under the interaction model?

Part 3: Perform an = 0:05 level test of the null hypothesis in (b). What do you conclude? Can you perform this test under the main effect model? If yes, carry out the test. If not, explain why.

Using the following DATASET

rat caps so2 nn 1 0 1 0.371 2 1 1 0.696 3 0 1 0.359 4 1 1 0.645 5 1 1 0.662 6 0 0 0.227 7 1 0 0.442 8 1 1 0.722 9 1 0 0.416 10 0 1 0.399 11 1 1 0.745 12 0 1 0.277 13 1 0 0.522 14 1 0 0.363 15 0 0 0.271 16 1 1 0.698 17 0 0 0.12 18 0 0 0.239 19 0 1 0.336 20 1 1 0.62 21 1 1 0.743 22 1 0 0.444 23 1 0 0.533 24 1 1 0.673 25 1 0 0.507 26 1 0 0.389 27 1 0 0.389 28 1 0 0.549 29 0 1 0.312 30 0 0 0.289 31 1 1 0.699 32 1 1 0.718 33 1 0 0.581 34 1 1 0.641 35 0 0 0.202 36 0 1 0.323 37 1 1 0.686 38 0 0 0.256 39 0 1 0.432 40 0 0 0.191 41 0 0 0.35 42 0 1 0.33 43 0 1 0.389 44 0 1 0.297 45 0 0 0.215 46 0 0 0.279 47 0 0 0.135 48 0 1 0.338 49 0 1 0.32 50 1 1 0.694 51 0 0 0.33 52 0 1 0.299 53 1 1 0.795 54 0 0 0.275 55 1 0 0.517 56 0 0 0.308 57 0 0 0.211 58 1 1 0.66 59 1 1 0.673 60 0 1 0.38 61 1 0 0.445 62 0 0 0.176 63 0 1 0.459 64 0 1 0.285 65 1 1 0.71 66 0 1 0.334 67 1 1 0.631 68 1 1 0.723 69 1 1 0.748 70 0 1 0.34 71 1 1 0.656 72 1 0 0.457 73 1 0 0.384 74 0 0 0.258 75 0 1 0.317 76 0 0 0.256 77 0 1 0.326 78 1 0 0.535 79 0 1 0.348 80 0 0 0.375 81 0 0 0.254 82 1 0 0.478 83 0 1 0.291 84 1 1 0.711 85 0 0 0.191 86 0 1 0.367 87 1 1 0.648 88 1 1 0.722 89 1 1 0.681 90 0 0 0.161 91 0 0 0.301 92 1 0 0.513 93 0 1 0.282 94 1 0 0.472 95 0 1 0.291 96 0 0 0.209 97 1 1 0.725 98 0 0 0.284 99 0 1 0.386 100 1 0 0.515 101 1 0 0.473 102 0 1 0.174 103 0 1 0.367 104 1 1 0.745 105 0 1 0.364 106 1 1 0.64 107 1 1 0.723 108 1 0 0.529 109 0 1 0.266 110 0 0 0.266 111 1 1 0.823 112 0 1 0.258 113 0 0 0.28 114 1 1 0.683 115 1 0 0.41 116 0 0 0.194 117 1 0 0.48 118 1 1 0.68 119 1 0 0.411 120 0 1 0.327 121 1 0 0.433 122 0 1 0.377 123 0 1 0.385 124 0 0 0.223 125 0 1 0.251 126 1 1 0.723 127 0 0 0.142 128 0 0 0.214 129 0 0 0.264 130 1 0 0.525 131 1 0 0.512 132 1 1 0.748 133 1 0 0.45 134 1 0 0.413 135 1 0 0.45 136 1 0 0.57 137 1 1 0.753 138 1 1 0.685 139 1 1 0.659 140 0 1 0.401 141 0 1 0.351 142 0 1 0.344 143 0 0 0.133 144 0 0 0.314 145 1 1 0.658 146 0 0 0.26 147 0 0 0.284 148 0 0 0.3 149 0 0 0.306 150 1 0 0.47 151 1 1 0.728 152 0 1 0.236 153 0 1 0.339 154 0 0 0.284 155 0 0 0.223 156 0 1 0.288 157 1 0 0.502 158 0 0 0.25 159 0 1 0.263 160 0 0 0.181 161 1 0 0.539 162 0 1 0.319 163 1 0 0.387 164 0 1 0.364 165 0 1 0.316 166 1 0 0.522 167 1 0 0.526 168 0 0 0.224 169 0 1 0.275 170 0 1 0.463 171 1 0 0.519 172 0 0 0.274 173 1 1 0.615 174 1 0 0.535 175 1 0 0.464 176 1 0 0.463 177 0 1 0.392 178 1 0 0.454 179 1 1 0.751 180 1 0 0.511 181 1 1 0.661 182 0 1 0.374 183 1 0 0.47 184 0 1 0.382 185 1 0 0.465 186 0 1 0.356 187 0 1 0.272 188 1 0 0.495 189 1 1 0.769 190 1 1 0.708 191 0 1 0.393 192 0 1 0.261 193 1 1 0.681 194 1 0 0.523 195 1 1 0.728 196 0 1 0.268 197 0 1 0.29 198 1 0 0.433 199 0 0 0.243 200 1 0 0.451 201 0 0 0.251 202 1 1 0.658 203 1 0 0.473 204 1 0 0.508 205 0 1 0.4 206 1 1 0.706 207 0 1 0.292 208 0 1 0.307 209 0 0 0.257 210 0 0 0.186 211 0 1 0.35 212 0 1 0.275 213 0 0 0.222 214 1 0 0.448 215 0 0 0.295 216 1 0 0.497 217 0 1 0.324 218 0 1 0.331 219 1 1 0.703 220 1 1 0.731 221 0 1 0.325 222 1 1 0.566 223 1 0 0.418 224 1 0 0.509 225 0 0 0.297 226 0 1 0.375 227 0 1 0.354 228 1 0 0.38 229 1 0 0.537 230 1 1 0.772 231 1 0 0.456 232 0 1 0.454 233 1 0 0.328 234 0 0 0.188 235 0 1 0.352 236 0 1 0.344 237 0 0 0.166 238 1 1 0.782 239 0 0 0.206 240 1 1 0.627 241 1 0 0.479 242 1 0 0.541 243 0 0 0.199 244 1 0 0.444 245 0 1 0.338 246 1 0 0.427 247 0 1 0.329 248 1 1 0.72 249 1 1 0.744 250 1 0 0.523 251 0 1 0.306 252 0 0 0.228 253 1 0 0.524 254 0 1 0.386 255 1 0 0.5 256 1 1 0.695 257 0 0 0.136 258 0 1 0.409 259 0 1 0.345 260 1 1 0.699 261 1 1 0.688 262 1 1 0.671 263 1 0 0.448 264 1 1 0.683 265 1 0 0.396 266 1 0 0.573 267 0 1 0.337 268 0 1 0.362 269 0 0 0.266 270 1 0 0.54 271 1 1 0.646 272 0 0 0.183 273 0 0 0.248 274 0 0 0.215 275 1 1 0.818

Explanation / Answer

Using main effects:

> summary(lm(nn ~ caps*so2, data = X125))

Call:
lm(formula = nn ~ caps * so2, data = X125)

Residuals:
Min 1Q Median 3Q Max
-0.161885 -0.037403 0.000855 0.039135 0.136855

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.238145 0.006741 35.326 <2e-16 ***
caps 0.237700 0.009227 25.762 <2e-16 ***
so2 0.097739 0.009032 10.822 <2e-16 ***
caps:so2 0.126337 0.012856 9.827 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.05308 on 271 degrees of freedom
Multiple R-squared: 0.9106,   Adjusted R-squared: 0.9096
F-statistic: 920.3 on 3 and 271 DF, p-value: < 2.2e-16

part B)

> anova(df)
Analysis of Variance Table

Response: nn
Df Sum Sq Mean Sq F value Pr(>F)
caps 1 5.7590 5.7590 2043.91 < 2.2e-16 ***
so2 1 1.7479 1.7479 620.35 < 2.2e-16 ***
caps:so2 1 0.2721 0.2721 96.57 < 2.2e-16 ***
Residuals 271 0.7636 0.0028
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Part c)

F statistic is significant and we can reject null hypothesis on main effect model.

ii) Regression of interaction model:

> df1 <- lm(nn ~ caps+ so2, data = X125)
> summary(df1)

Call:
lm(formula = nn ~ caps + so2, data = X125)

Residuals:
Min 1Q Median 3Q Max
-0.189497 -0.039997 0.003503 0.039662 0.171593

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.203407 0.006673 30.48 <2e-16 ***
caps 0.302773 0.007469 40.54 <2e-16 ***
so2 0.160090 0.007472 21.43 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.06171 on 272 degrees of freedom
Multiple R-squared: 0.8788,   Adjusted R-squared: 0.8779
F-statistic: 985.8 on 2 and 272 DF, p-value: < 2.2e-16

Part B)

> anova(df1)
Analysis of Variance Table

Response: nn
Df Sum Sq Mean Sq F value Pr(>F)
caps 1 5.7590 5.7590 1512.48 < 2.2e-16 ***
so2 1 1.7479 1.7479 459.06 < 2.2e-16 ***
Residuals 272 1.0357 0.0038
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Part C)

At 0.05 significant level both are significant. we can reject null hypothesis. variance between variables is different.

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