This tutorial draws aims at making various binary outcome GLM models interpretable through the use of plots. As such, it begins by setting up some data (involving a few covariates) and then generating various versions of an outcome based upon data-generating proceses with and without interaction. The aim of the tutorial is to both highlight the use of predicted probability plots for demonstrating effects and demonstrate the challenge - even then - of clearly communicating the results of these types of models.
Let's begin by generating our covariates:
set.seed(1)
n <- 200
x1 <- rbinom(n, 1, 0.5)
x2 <- runif(n, 0, 1)
x3 <- runif(n, 0, 5)
Now, we'll build several models. Each model has an outcome that is a transformed linear function the covariates (i.e., we calculate a y variable that is a linear function of the covariates, then rescale that outcome [0,1], and use the rescaled version as a probability model in generating draws from a binomial distribution).
# Simple multivariate model (no interaction):
y1 <- 2 * x1 + 5 * x2 + rnorm(n, 0, 3)
y1s <- rbinom(n, 1, (y1 - min(y1))/(max(y1) - min(y1))) # the math here is just to rescale to [0,1]
# Simple multivariate model (with interaction):
y2 <- 2 * x1 + 5 * x2 + 2 * x1 * x2 + rnorm(n, 0, 3)
y2s <- rbinom(n, 1, (y2 - min(y2))/(max(y2) - min(y2)))
# Simple multivariate model (with interaction and an extra term):
y3 <- 2 * x1 + 5 * x2 + 2 * x1 * x2 + x3 + rnorm(n, 0, 3)
y3s <- rbinom(n, 1, (y2 - min(y2))/(max(y2) - min(y2)))
We thus have three outcomes (y1s, y2s, and y3s) that are binary outcomes, but each is constructed as a slightly different function of our three covariates.
We can then build models of each outcome. We'll build two versions of y2s and y3s (one version a that does not model the interaction and another version b that does model it):
m1 <- glm(y1s ~ x1 + x2, family = binomial(link = "probit"))
m2a <- glm(y2s ~ x1 + x2, family = binomial(link = "probit"))
m2b <- glm(y2s ~ x1 * x2, family = binomial(link = "probit"))
m3a <- glm(y1s ~ x1 + x2 + x3, family = binomial(link = "probit"))
m3b <- glm(y1s ~ x1 * x2 + x3, family = binomial(link = "probit"))
We can look at the outcome of one of our models, e.g. m3b (for y3s modelled with an interaction), but we know that the coefficients are not directly interpretable:
summary(m3b)
##
## Call:
## glm(formula = y1s ~ x1 * x2 + x3, family = binomial(link = "probit"))
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.554 -1.141 0.873 1.011 1.365
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -0.1591 0.2769 -0.57 0.57
## x1 0.4963 0.3496 1.42 0.16
## x2 0.3212 0.4468 0.72 0.47
## x3 -0.0315 0.0625 -0.50 0.61
## x1:x2 -0.0794 0.6429 -0.12 0.90
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 274.83 on 199 degrees of freedom
## Residual deviance: 266.59 on 195 degrees of freedom
## AIC: 276.6
##
## Number of Fisher Scoring iterations: 4
Instead we need to look at fitted values (specifically, the predicted probability of observing y==1 in each model. We can see these fitted values for our actual data using the predict function:
p3b.fitted <- predict(m3b, type = "response", se.fit = TRUE)
p3b.fitted
## $fit
## 1 2 3 4 5 6 7 8 9 10
## 0.4298 0.4530 0.6225 0.6029 0.4015 0.6364 0.6609 0.5970 0.6545 0.4695
## 11 12 13 14 15 16 17 18 19 20
## 0.4973 0.4273 0.6569 0.5082 0.6642 0.4463 0.6614 0.6982 0.5151 0.6141
## 21 22 23 24 25 26 27 28 29 30
## 0.6302 0.4258 0.6220 0.4929 0.5332 0.4765 0.4642 0.3856 0.6203 0.4908
## 31 32 33 34 35 36 37 38 39 40
## 0.4228 0.6397 0.4608 0.4837 0.6609 0.6472 0.6508 0.5043 0.6425 0.4370
## 41 42 43 44 45 46 47 48 49 50
## 0.6572 0.6667 0.6839 0.6086 0.6539 0.6261 0.4685 0.3956 0.6660 0.7103
## 51 52 53 54 55 56 57 58 59 60
## 0.4737 0.6890 0.4736 0.5032 0.4953 0.4100 0.4323 0.6188 0.6302 0.5521
## 61 62 63 64 65 66 67 68 69 70
## 0.6694 0.4475 0.4720 0.5097 0.6794 0.3998 0.4009 0.6893 0.4861 0.6063
## 71 72 73 74 75 76 77 78 79 80
## 0.4162 0.5845 0.4409 0.4336 0.4674 0.6353 0.6039 0.4327 0.6459 0.6608
## 81 82 83 84 85 86 87 88 89 90
## 0.3937 0.6701 0.4960 0.4253 0.6011 0.4232 0.6499 0.4563 0.3995 0.4565
## 91 92 93 94 95 96 97 98 99 100
## 0.4641 0.3956 0.7014 0.6519 0.6543 0.6663 0.4225 0.4199 0.5856 0.7099
## 101 102 103 104 105 106 107 108 109 110
## 0.6602 0.4064 0.4584 0.6347 0.6382 0.5027 0.4342 0.4874 0.5928 0.6369
## 111 112 113 114 115 116 117 118 119 120
## 0.5924 0.6219 0.5437 0.4831 0.4262 0.4882 0.6009 0.4480 0.5396 0.6509
## 121 122 123 124 125 126 127 128 129 130
## 0.6434 0.4324 0.4667 0.5559 0.6854 0.5004 0.6580 0.4891 0.4115 0.6425
## 131 132 133 134 135 136 137 138 139 140
## 0.6845 0.4723 0.4666 0.6879 0.6321 0.6621 0.6502 0.6263 0.6739 0.6822
## 141 142 143 144 145 146 147 148 149 150
## 0.6963 0.5908 0.4650 0.4652 0.6807 0.4638 0.4344 0.6252 0.4487 0.6544
## 151 152 153 154 155 156 157 158 159 160
## 0.6247 0.6036 0.5363 0.4336 0.6748 0.4489 0.7010 0.4441 0.4986 0.4451
## 161 162 163 164 165 166 167 168 169 170
## 0.4102 0.6543 0.5224 0.6137 0.6317 0.5000 0.4308 0.4861 0.6780 0.4361
## 171 172 173 174 175 176 177 178 179 180
## 0.6620 0.6581 0.6992 0.4602 0.4137 0.6391 0.6228 0.6851 0.5977 0.6332
## 181 182 183 184 185 186 187 188 189 190
## 0.4698 0.4494 0.6532 0.7151 0.6177 0.4415 0.6977 0.6755 0.5844 0.6767
## 191 192 193 194 195 196 197 198 199 200
## 0.6125 0.4399 0.5062 0.6395 0.4052 0.6253 0.4909 0.6311 0.4578 0.6187
##
## $se.fit
## 1 2 3 4 5 6 7 8 9
## 0.06090 0.07493 0.07213 0.07595 0.08505 0.05446 0.05024 0.08497 0.08736
## 10 11 12 13 14 15 16 17 18
## 0.08458 0.11678 0.07902 0.07268 0.10658 0.07666 0.05544 0.05338 0.08647
## 19 20 21 22 23 24 25 26 27
## 0.10526 0.06544 0.06385 0.06895 0.05878 0.07211 0.10434 0.05489 0.07976
## 28 29 30 31 32 33 34 35 36
## 0.09837 0.06283 0.09679 0.07274 0.09728 0.05479 0.06119 0.06930 0.06898
## 37 38 39 40 41 42 43 44 45
## 0.06506 0.08250 0.04903 0.06905 0.08040 0.05357 0.08155 0.07871 0.05722
## 46 47 48 49 50 51 52 53 54
## 0.07044 0.05457 0.08967 0.06453 0.09121 0.09065 0.08373 0.05496 0.07556
## 55 56 57 58 59 60 61 62 63
## 0.07928 0.07808 0.06067 0.07489 0.05318 0.12015 0.06053 0.05486 0.08607
## 64 65 66 67 68 69 70 71 72
## 0.08180 0.06173 0.08619 0.08555 0.07004 0.06089 0.07659 0.08207 0.09648
## 73 74 75 76 77 78 79 80 81
## 0.05605 0.06985 0.07491 0.07977 0.07469 0.06774 0.04755 0.07062 0.09192
## 82 83 84 85 86 87 88 89 90
## 0.06098 0.09897 0.07266 0.09254 0.07215 0.06856 0.09395 0.08548 0.07833
## 91 92 93 94 95 96 97 98 99
## 0.08700 0.08957 0.09002 0.06974 0.04870 0.05586 0.07756 0.07460 0.09843
## 100 101 102 103 104 105 106 107 108
## 0.09045 0.05616 0.08076 0.05338 0.05133 0.05233 0.11964 0.06895 0.09019
## 109 110 111 112 113 114 115 116 117
## 0.09251 0.05045 0.08785 0.07194 0.11253 0.07388 0.06328 0.06781 0.08778
## 118 119 120 121 122 123 124 125 126
## 0.05248 0.11138 0.05489 0.06888 0.05946 0.05557 0.12468 0.07418 0.11247
## 127 128 129 130 131 132 133 134 135
## 0.08133 0.08255 0.07904 0.09024 0.09377 0.08286 0.05628 0.07110 0.10181
## 136 137 138 139 140 141 142 143 144
## 0.07580 0.05096 0.08026 0.06026 0.09215 0.09064 0.08885 0.05537 0.05696
## 145 146 147 148 149 150 151 152 153
## 0.06283 0.07662 0.07562 0.07680 0.05194 0.06802 0.06453 0.09005 0.10586
## 154 155 156 157 158 159 160 161 162
## 0.06816 0.09976 0.05952 0.08116 0.09688 0.09872 0.06212 0.07796 0.07138
## 163 164 165 166 167 168 169 170 171
## 0.09198 0.06561 0.07026 0.08818 0.08680 0.06974 0.06071 0.07222 0.06292
## 172 173 174 175 176 177 178 179 180
## 0.04906 0.08438 0.08074 0.07343 0.04938 0.05798 0.07527 0.08185 0.05499
## 181 182 183 184 185 186 187 188 189
## 0.05549 0.07398 0.05759 0.09607 0.06677 0.06207 0.07768 0.07232 0.09640
## 190 191 192 193 194 195 196 197 198
## 0.09458 0.07770 0.07854 0.10283 0.05069 0.08024 0.05617 0.06491 0.05762
## 199 200
## 0.05557 0.07847
##
## $residual.scale
## [1] 1
We can even draw a small plot showing the predicted values separately for levels of x1 (recall that x1 is a binary/indicator variable):
plot(NA, xlim = c(0, 1), ylim = c(0, 1), xlab = "x2", ylab = "Predicted Probability of y=1")
points(x2[x1 == 0], p3b.fitted$fit[x1 == 0], col = rgb(1, 0, 0, 0.5))
points(x2[x1 == 1], p3b.fitted$fit[x1 == 1], col = rgb(0, 0, 1, 0.5))
But this graph doesn't show the fit of the model to all values of x1 and x2 (or x3) and doesn't communicate any of our uncertainty.
To get a better grasp on our models, we'll create some fake data representing the full scales of x1, x2, and x3:
newdata1 <- expand.grid(x1 = 0:1, x2 = seq(0, 1, length.out = 10))
newdata2 <- expand.grid(x1 = 0:1, x2 = seq(0, 1, length.out = 10), x3 = seq(0,
5, length.out = 25))
We can then use these new fake data to generate predicted probabilities of each outcome at each combination of covarites:
p1 <- predict(m1, newdata1, type = "response", se.fit = TRUE)
p2a <- predict(m2a, newdata1, type = "response", se.fit = TRUE)
p2b <- predict(m2b, newdata1, type = "response", se.fit = TRUE)
p3a <- predict(m3a, newdata2, type = "response", se.fit = TRUE)
p3b <- predict(m3b, newdata2, type = "response", se.fit = TRUE)
We can look at one of these objects, e.g. p3b, to see that we have predicted probabilities and associated standard errors:
p3b
## $fit
## 1 2 3 4 5 6 7 8 9 10
## 0.4368 0.6320 0.4509 0.6421 0.4650 0.6521 0.4792 0.6619 0.4935 0.6717
## 11 12 13 14 15 16 17 18 19 20
## 0.5077 0.6814 0.5219 0.6909 0.5361 0.7003 0.5503 0.7096 0.5644 0.7187
## 21 22 23 24 25 26 27 28 29 30
## 0.4342 0.6295 0.4483 0.6396 0.4624 0.6496 0.4766 0.6595 0.4909 0.6693
## 31 32 33 34 35 36 37 38 39 40
## 0.5051 0.6790 0.5193 0.6886 0.5335 0.6980 0.5477 0.7073 0.5618 0.7165
## 41 42 43 44 45 46 47 48 49 50
## 0.4316 0.6271 0.4457 0.6372 0.4598 0.6472 0.4740 0.6571 0.4882 0.6670
## 51 52 53 54 55 56 57 58 59 60
## 0.5025 0.6767 0.5167 0.6863 0.5309 0.6957 0.5451 0.7051 0.5592 0.7143
## 61 62 63 64 65 66 67 68 69 70
## 0.4291 0.6246 0.4431 0.6347 0.4572 0.6448 0.4714 0.6547 0.4856 0.6646
## 71 72 73 74 75 76 77 78 79 80
## 0.4999 0.6743 0.5141 0.6839 0.5283 0.6934 0.5425 0.7028 0.5566 0.7120
## 81 82 83 84 85 86 87 88 89 90
## 0.4265 0.6221 0.4405 0.6323 0.4546 0.6423 0.4688 0.6523 0.4830 0.6622
## 91 92 93 94 95 96 97 98 99 100
## 0.4972 0.6719 0.5115 0.6816 0.5257 0.6911 0.5399 0.7005 0.5540 0.7098
## 101 102 103 104 105 106 107 108 109 110
## 0.4239 0.6196 0.4379 0.6298 0.4520 0.6399 0.4662 0.6499 0.4804 0.6598
## 111 112 113 114 115 116 117 118 119 120
## 0.4946 0.6696 0.5089 0.6792 0.5231 0.6888 0.5373 0.6982 0.5514 0.7075
## 121 122 123 124 125 126 127 128 129 130
## 0.4213 0.6171 0.4354 0.6273 0.4494 0.6374 0.4636 0.6475 0.4778 0.6574
## 131 132 133 134 135 136 137 138 139 140
## 0.4920 0.6672 0.5062 0.6769 0.5205 0.6865 0.5347 0.6959 0.5488 0.7053
## 141 142 143 144 145 146 147 148 149 150
## 0.4188 0.6146 0.4328 0.6248 0.4468 0.6350 0.4610 0.6450 0.4752 0.6550
## 151 152 153 154 155 156 157 158 159 160
## 0.4894 0.6648 0.5036 0.6745 0.5179 0.6842 0.5321 0.6936 0.5462 0.7030
## 161 162 163 164 165 166 167 168 169 170
## 0.4162 0.6121 0.4302 0.6223 0.4443 0.6325 0.4584 0.6426 0.4726 0.6525
## 171 172 173 174 175 176 177 178 179 180
## 0.4868 0.6624 0.5010 0.6722 0.5153 0.6818 0.5295 0.6913 0.5436 0.7007
## 181 182 183 184 185 186 187 188 189 190
## 0.4137 0.6096 0.4276 0.6198 0.4417 0.6300 0.4558 0.6401 0.4700 0.6501
## 191 192 193 194 195 196 197 198 199 200
## 0.4842 0.6600 0.4984 0.6698 0.5126 0.6795 0.5269 0.6890 0.5410 0.6985
## 201 202 203 204 205 206 207 208 209 210
## 0.4111 0.6071 0.4251 0.6173 0.4391 0.6275 0.4532 0.6377 0.4674 0.6477
## 211 212 213 214 215 216 217 218 219 220
## 0.4816 0.6576 0.4958 0.6674 0.5100 0.6771 0.5242 0.6867 0.5384 0.6962
## 221 222 223 224 225 226 227 228 229 230
## 0.4086 0.6045 0.4225 0.6148 0.4365 0.6251 0.4506 0.6352 0.4647 0.6453
## 231 232 233 234 235 236 237 238 239 240
## 0.4789 0.6552 0.4932 0.6650 0.5074 0.6748 0.5216 0.6844 0.5358 0.6939
## 241 242 243 244 245 246 247 248 249 250
## 0.4060 0.6020 0.4199 0.6123 0.4339 0.6226 0.4480 0.6327 0.4621 0.6428
## 251 252 253 254 255 256 257 258 259 260
## 0.4763 0.6528 0.4906 0.6627 0.5048 0.6724 0.5190 0.6821 0.5332 0.6916
## 261 262 263 264 265 266 267 268 269 270
## 0.4035 0.5995 0.4174 0.6098 0.4313 0.6201 0.4454 0.6303 0.4595 0.6404
## 271 272 273 274 275 276 277 278 279 280
## 0.4737 0.6504 0.4879 0.6603 0.5022 0.6700 0.5164 0.6797 0.5306 0.6893
## 281 282 283 284 285 286 287 288 289 290
## 0.4010 0.5969 0.4148 0.6073 0.4288 0.6176 0.4428 0.6278 0.4569 0.6379
## 291 292 293 294 295 296 297 298 299 300
## 0.4711 0.6479 0.4853 0.6579 0.4996 0.6677 0.5138 0.6774 0.5280 0.6869
## 301 302 303 304 305 306 307 308 309 310
## 0.3984 0.5944 0.4123 0.6048 0.4262 0.6151 0.4402 0.6253 0.4543 0.6354
## 311 312 313 314 315 316 317 318 319 320
## 0.4685 0.6455 0.4827 0.6554 0.4970 0.6653 0.5112 0.6750 0.5254 0.6846
## 321 322 323 324 325 326 327 328 329 330
## 0.3959 0.5919 0.4097 0.6023 0.4236 0.6126 0.4376 0.6228 0.4517 0.6330
## 331 332 333 334 335 336 337 338 339 340
## 0.4659 0.6431 0.4801 0.6530 0.4943 0.6629 0.5086 0.6726 0.5228 0.6823
## 341 342 343 344 345 346 347 348 349 350
## 0.3934 0.5893 0.4072 0.5997 0.4211 0.6101 0.4351 0.6203 0.4492 0.6305
## 351 352 353 354 355 356 357 358 359 360
## 0.4633 0.6406 0.4775 0.6506 0.4917 0.6605 0.5060 0.6703 0.5202 0.6799
## 361 362 363 364 365 366 367 368 369 370
## 0.3909 0.5868 0.4046 0.5972 0.4185 0.6075 0.4325 0.6178 0.4466 0.6280
## 371 372 373 374 375 376 377 378 379 380
## 0.4607 0.6382 0.4749 0.6482 0.4891 0.6581 0.5033 0.6679 0.5176 0.6776
## 381 382 383 384 385 386 387 388 389 390
## 0.3883 0.5842 0.4021 0.5946 0.4159 0.6050 0.4299 0.6153 0.4440 0.6256
## 391 392 393 394 395 396 397 398 399 400
## 0.4581 0.6357 0.4723 0.6457 0.4865 0.6557 0.5007 0.6655 0.5150 0.6752
## 401 402 403 404 405 406 407 408 409 410
## 0.3858 0.5816 0.3995 0.5921 0.4134 0.6025 0.4273 0.6128 0.4414 0.6231
## 411 412 413 414 415 416 417 418 419 420
## 0.4555 0.6332 0.4697 0.6433 0.4839 0.6533 0.4981 0.6631 0.5123 0.6729
## 421 422 423 424 425 426 427 428 429 430
## 0.3833 0.5791 0.3970 0.5896 0.4108 0.6000 0.4248 0.6103 0.4388 0.6206
## 431 432 433 434 435 436 437 438 439 440
## 0.4529 0.6308 0.4671 0.6408 0.4813 0.6508 0.4955 0.6607 0.5097 0.6705
## 441 442 443 444 445 446 447 448 449 450
## 0.3808 0.5765 0.3945 0.5870 0.4083 0.5974 0.4222 0.6078 0.4362 0.6181
## 451 452 453 454 455 456 457 458 459 460
## 0.4503 0.6283 0.4645 0.6384 0.4787 0.6484 0.4929 0.6583 0.5071 0.6681
## 461 462 463 464 465 466 467 468 469 470
## 0.3783 0.5739 0.3920 0.5845 0.4057 0.5949 0.4196 0.6053 0.4336 0.6156
## 471 472 473 474 475 476 477 478 479 480
## 0.4477 0.6258 0.4619 0.6359 0.4760 0.6460 0.4903 0.6559 0.5045 0.6657
## 481 482 483 484 485 486 487 488 489 490
## 0.3758 0.5714 0.3895 0.5819 0.4032 0.5924 0.4171 0.6027 0.4311 0.6131
## 491 492 493 494 495 496 497 498 499 500
## 0.4451 0.6233 0.4593 0.6335 0.4734 0.6435 0.4877 0.6535 0.5019 0.6634
##
## $se.fit
## 1 2 3 4 5 6 7 8 9
## 0.10906 0.11971 0.09785 0.10469 0.08926 0.09169 0.08427 0.08147 0.08364
## 10 11 12 13 14 15 16 17 18
## 0.07488 0.08750 0.07263 0.09527 0.07477 0.10602 0.08065 0.11883 0.08923
## 19 20 21 22 23 24 25 26 27
## 0.13296 0.09953 0.10626 0.11718 0.09465 0.10190 0.08564 0.08862 0.08033
## 28 29 30 31 32 33 34 35 36
## 0.07816 0.07956 0.07148 0.08352 0.06934 0.09157 0.07183 0.10267 0.07818
## 37 38 39 40 41 42 43 44 45
## 0.11584 0.08725 0.13030 0.09800 0.10365 0.11478 0.09163 0.09923 0.08220
## 46 47 48 49 50 51 52 53 54
## 0.08568 0.07653 0.07498 0.07562 0.06818 0.07968 0.06617 0.08801 0.06903
## 55 56 57 58 59 60 61 62 63
## 0.09947 0.07586 0.11299 0.08542 0.12779 0.09661 0.10123 0.11251 0.08881
## 64 65 66 67 68 69 70 71 72
## 0.09671 0.07895 0.08288 0.07291 0.07193 0.07184 0.06502 0.07599 0.06315
## 73 74 75 76 77 78 79 80 81
## 0.08461 0.06638 0.09642 0.07372 0.11030 0.08376 0.12542 0.09538 0.09903
## 82 83 84 85 86 87 88 89 90
## 0.11040 0.08621 0.09435 0.07591 0.08025 0.06949 0.06905 0.06824 0.06203
## 91 92 93 94 95 96 97 98 99
## 0.07249 0.06030 0.08139 0.06393 0.09355 0.07176 0.10777 0.08229 0.12320
## 100 101 102 103 104 105 106 107 108
## 0.09432 0.09705 0.10845 0.08386 0.09217 0.07312 0.07780 0.06631 0.06636
## 109 110 111 112 113 114 115 116 117
## 0.06485 0.05922 0.06920 0.05766 0.07838 0.06170 0.09088 0.07003 0.10543
## 118 119 120 121 122 123 124 125 126
## 0.08102 0.12115 0.09344 0.09530 0.10667 0.08176 0.09017 0.07060 0.07556
## 127 128 129 130 131 132 133 134 135
## 0.06339 0.06388 0.06173 0.05665 0.06614 0.05525 0.07560 0.05971 0.08843
## 136 137 138 139 140 141 142 143 144
## 0.06853 0.10328 0.07997 0.11927 0.09276 0.09381 0.10508 0.07994 0.08839
## 145 146 147 148 149 150 151 152 153
## 0.06838 0.07355 0.06077 0.06166 0.05889 0.05435 0.06337 0.05313 0.07307
## 154 155 156 157 158 159 160 161 162
## 0.05801 0.08621 0.06730 0.10134 0.07914 0.11758 0.09227 0.09257 0.10369
## 163 164 165 166 167 168 169 170 171
## 0.07842 0.08683 0.06650 0.07179 0.05850 0.05973 0.05639 0.05236 0.06091
## 172 173 174 175 176 177 178 179 180
## 0.05134 0.07084 0.05662 0.08424 0.06635 0.09963 0.07856 0.11608 0.09198
## 181 182 183 184 185 186 187 188 189
## 0.09160 0.10251 0.07722 0.08551 0.06497 0.07032 0.05662 0.05811 0.05427
## 190 191 192 193 194 195 196 197 198
## 0.05072 0.05881 0.04992 0.06892 0.05558 0.08255 0.06570 0.09815 0.07823
## 199 200 201 202 203 204 205 206 207
## 0.11479 0.09191 0.09091 0.10154 0.07633 0.08445 0.06382 0.06915 0.05515
## 208 209 210 211 212 213 214 215 216
## 0.05686 0.05259 0.04949 0.05711 0.04891 0.06735 0.05491 0.08115 0.06536
## 217 218 219 220 221 222 223 224 225
## 0.09691 0.07817 0.11370 0.09206 0.09049 0.10081 0.07578 0.08366 0.06306
## 226 227 228 229 230 231 232 233 234
## 0.06831 0.05415 0.05599 0.05137 0.04869 0.05583 0.04833 0.06615 0.05464
## 235 236 237 238 239 240 241 242 243
## 0.08006 0.06536 0.09594 0.07837 0.11284 0.09243 0.09035 0.10031 0.07557
## 244 245 246 247 248 249 250 251 252
## 0.08315 0.06272 0.06780 0.05362 0.05553 0.05065 0.04836 0.05502 0.04823
## 253 254 255 256 257 258 259 260 261
## 0.06533 0.05477 0.07929 0.06568 0.09524 0.07885 0.11220 0.09303 0.09049
## 262 263 264 265 266 267 268 269 270
## 0.10005 0.07570 0.08293 0.06280 0.06765 0.05358 0.05549 0.05045 0.04852
## 271 272 273 274 275 276 277 278 279
## 0.05468 0.04860 0.06492 0.05532 0.07886 0.06634 0.09480 0.07959 0.11180
## 280 281 282 283 284 285 286 287 288
## 0.09384 0.09091 0.10004 0.07617 0.08301 0.06328 0.06785 0.05403 0.05589
## 289 290 291 292 293 294 295 296 297
## 0.05078 0.04916 0.05483 0.04944 0.06492 0.05627 0.07876 0.06733 0.09465
## 298 299 300 301 302 303 304 305 306
## 0.08061 0.11162 0.09488 0.09159 0.10028 0.07695 0.08338 0.06417 0.06842
## 307 308 309 310 311 312 313 314 315
## 0.05496 0.05671 0.05163 0.05027 0.05547 0.05074 0.06533 0.05761 0.07900
## 316 317 318 319 320 321 322 323 324
## 0.06864 0.09478 0.08188 0.11168 0.09614 0.09252 0.10077 0.07806 0.08406
## 325 326 327 328 329 330 331 332 333
## 0.06543 0.06934 0.05633 0.05796 0.05295 0.05183 0.05657 0.05247 0.06615
## 334 335 336 337 338 339 340 341 342
## 0.05932 0.07957 0.07026 0.09519 0.08342 0.11198 0.09762 0.09371 0.10151
## 343 344 345 346 347 348 349 350 351
## 0.07945 0.08502 0.06705 0.07061 0.05812 0.05959 0.05473 0.05380 0.05811
## 352 353 354 355 356 357 358 359 360
## 0.05459 0.06735 0.06138 0.08048 0.07218 0.09587 0.08519 0.11251 0.09930
## 361 362 363 364 365 366 367 368 369
## 0.09513 0.10250 0.08113 0.08628 0.06900 0.07221 0.06028 0.06160 0.05691
## 370 371 372 373 374 375 376 377 378
## 0.05616 0.06004 0.05707 0.06891 0.06375 0.08170 0.07436 0.09682 0.08721
## 379 380 381 382 383 384 385 386 387
## 0.11328 0.10118 0.09678 0.10372 0.08306 0.08781 0.07124 0.07412 0.06277
## 388 389 390 391 392 393 394 395 396
## 0.06394 0.05945 0.05885 0.06234 0.05987 0.07082 0.06642 0.08322 0.07681
## 397 398 399 400 401 402 403 404 405
## 0.09804 0.08945 0.11426 0.10326 0.09863 0.10518 0.08524 0.08960 0.07374
## 406 407 408 409 410 411 412 413 414
## 0.07633 0.06555 0.06659 0.06230 0.06184 0.06496 0.06294 0.07304 0.06934
## 415 416 417 418 419 420 421 422 423
## 0.08503 0.07949 0.09951 0.09189 0.11548 0.10552 0.10067 0.10687 0.08762
## 424 425 426 427 428 429 430 431 432
## 0.09164 0.07649 0.07880 0.06858 0.06951 0.06541 0.06508 0.06787 0.06626
## 433 434 435 436 437 438 439 440 441
## 0.07554 0.07249 0.08710 0.08238 0.10122 0.09454 0.11690 0.10797 0.10289
## 442 443 444 445 446 447 448 449 450
## 0.10877 0.09021 0.09393 0.07944 0.08152 0.07183 0.07267 0.06875 0.06856
## 451 452 453 454 455 456 457 458 459
## 0.07101 0.06979 0.07829 0.07585 0.08942 0.08548 0.10316 0.09737 0.11853
## 460 461 462 463 464 465 466 467 468
## 0.11058 0.10528 0.11087 0.09297 0.09643 0.08258 0.08447 0.07527 0.07605
## 469 470 471 472 473 474 475 476 477
## 0.07228 0.07223 0.07437 0.07351 0.08127 0.07940 0.09197 0.08875 0.10531
## 478 479 480 481 482 483 484 485 486
## 0.10038 0.12036 0.11335 0.10782 0.11318 0.09588 0.09914 0.08587 0.08763
## 487 488 489 490 491 492 493 494 495
## 0.07886 0.07963 0.07598 0.07608 0.07791 0.07739 0.08445 0.08311 0.09473
## 496 497 498 499 500
## 0.09219 0.10767 0.10354 0.12238 0.11627
##
## $residual.scale
## [1] 1
It is then relatively straight forward to plot the predicted probabilities for all of our data. We'll start with the simple models, then look at the models with interactions and the additional covariate x3.
plot(NA, xlim = c(0, 1), ylim = c(0, 1), xlab = "x2", ylab = "Predicted Probability of y=1")
# `x1==0`
lines(newdata1$x2[newdata1$x1 == 0], p1$fit[newdata1$x1 == 0], col = "red")
lines(newdata1$x2[newdata1$x1 == 0], p1$fit[newdata1$x1 == 0] + 1.96 * p1$se.fit[newdata1$x1 ==
0], col = "red", lty = 2)
lines(newdata1$x2[newdata1$x1 == 0], p1$fit[newdata1$x1 == 0] - 1.96 * p1$se.fit[newdata1$x1 ==
0], col = "red", lty = 2)
# `x1==1`
lines(newdata1$x2[newdata1$x1 == 1], p1$fit[newdata1$x1 == 1], col = "blue")
lines(newdata1$x2[newdata1$x1 == 1], p1$fit[newdata1$x1 == 1] + 1.96 * p1$se.fit[newdata1$x1 ==
1], col = "blue", lty = 2)
lines(newdata1$x2[newdata1$x1 == 1], p1$fit[newdata1$x1 == 1] - 1.96 * p1$se.fit[newdata1$x1 ==
1], col = "blue", lty = 2)
The above plot shows two predicted probability curves with heavily overlapping confidence bands. While the effect of x2 is clearly different from zero for both x1==0 and x1==1, the difference between the two curves is not significant.
But this model is based on data with no underlying interaction. Let's look next at the outcome that is a function of an interaction between covariates.
Recall that the interaction model (with outcome y2s) was estimated in two different ways. The first estimated model did not account for the interaction, while the second estimated model did account for the interaction. Let's see the two models side-by-side to compare the inference we would draw about the interaction:
# Model estimated without interaction
layout(matrix(1:2, nrow = 1))
plot(NA, xlim = c(0, 1), ylim = c(0, 1), xlab = "x2", ylab = "Predicted Probability of y=1",
main = "Estimated without interaction")
# `x1==0`
lines(newdata1$x2[newdata1$x1 == 0], p2a$fit[newdata1$x1 == 0], col = "red")
lines(newdata1$x2[newdata1$x1 == 0], p2a$fit[newdata1$x1 == 0] + 1.96 * p2a$se.fit[newdata1$x1 ==
0], col = "red", lty = 2)
lines(newdata1$x2[newdata1$x1 == 0], p2a$fit[newdata1$x1 == 0] - 1.96 * p2a$se.fit[newdata1$x1 ==
0], col = "red", lty = 2)
# `x1==1`
lines(newdata1$x2[newdata1$x1 == 1], p2a$fit[newdata1$x1 == 1], col = "blue")
lines(newdata1$x2[newdata1$x1 == 1], p2a$fit[newdata1$x1 == 1] + 1.96 * p2a$se.fit[newdata1$x1 ==
1], col = "blue", lty = 2)
lines(newdata1$x2[newdata1$x1 == 1], p2a$fit[newdata1$x1 == 1] - 1.96 * p2a$se.fit[newdata1$x1 ==
1], col = "blue", lty = 2)
# Model estimated with interaction
plot(NA, xlim = c(0, 1), ylim = c(0, 1), xlab = "x2", ylab = "Predicted Probability of y=1",
main = "Estimated with interaction")
# `x1==0`
lines(newdata1$x2[newdata1$x1 == 0], p2b$fit[newdata1$x1 == 0], col = "red")
lines(newdata1$x2[newdata1$x1 == 0], p2b$fit[newdata1$x1 == 0] + 1.96 * p2b$se.fit[newdata1$x1 ==
0], col = "red", lty = 2)
lines(newdata1$x2[newdata1$x1 == 0], p2b$fit[newdata1$x1 == 0] - 1.96 * p2b$se.fit[newdata1$x1 ==
0], col = "red", lty = 2)
# `x1==1`
lines(newdata1$x2[newdata1$x1 == 1], p2b$fit[newdata1$x1 == 1], col = "blue")
lines(newdata1$x2[newdata1$x1 == 1], p2b$fit[newdata1$x1 == 1] + 1.96 * p2b$se.fit[newdata1$x1 ==
1], col = "blue", lty = 2)
lines(newdata1$x2[newdata1$x1 == 1], p2b$fit[newdata1$x1 == 1] - 1.96 * p2b$se.fit[newdata1$x1 ==
1], col = "blue", lty = 2)
The lefthand model leads us to some incorrect inference. Both predicted probability curves are essentially identical, suggesting that the influence of x2 is constant at both levels of x1. This is because our model did not account for any interaction.
The righthand model leads us to substantially different inference. When x1==0 (shown in red), there appears to be almost no effect of x2, but when x1==1, the effect of x2 is strongly positive.
When we add an additional covariate to the model, things become much more complicated. Recall that the predicted probabilities have to be calculated on some value of each covariate. In other words, we have to define the predicted probability in terms of all of the covariates in the model. Thus, when we add an additional covariate (even if it does not interact with our focal covariates x1 and x2), we need to account for it when estimating our predicted probabilities. We'll see this at work when we plot the predicted probabilities for our (incorrect) model estimated without the x1*x2 interaction and in our (correct) model estimated with that interaction.
No-Interaction model with an additional covariate
plot(NA, xlim = c(0, 1), ylim = c(0, 1), xlab = "x2", ylab = "Predicted Probability of y=1")
s <- sapply(unique(newdata2$x3), function(i) {
# `x1==0`
lines(newdata2$x2[newdata2$x1 == 0 & newdata2$x3 == i], p3a$fit[newdata2$x1 ==
0 & newdata2$x3 == i], col = rgb(1, 0, 0, 0.5))
lines(newdata2$x2[newdata2$x1 == 0 & newdata2$x3 == i], p3a$fit[newdata2$x1 ==
0 & newdata2$x3 == i] + 1.96 * p3a$se.fit[newdata2$x1 == 0 & newdata2$x3 ==
i], col = rgb(1, 0, 0, 0.5), lty = 2)
lines(newdata2$x2[newdata2$x1 == 0 & newdata2$x3 == i], p3a$fit[newdata2$x1 ==
0 & newdata2$x3 == i] - 1.96 * p3a$se.fit[newdata2$x1 == 0 & newdata2$x3 ==
i], col = rgb(1, 0, 0, 0.5), lty = 2)
# `x1==1`
lines(newdata2$x2[newdata2$x1 == 1 & newdata2$x3 == i], p3a$fit[newdata2$x1 ==
1 & newdata2$x3 == i], col = rgb(0, 0, 1, 0.5))
lines(newdata2$x2[newdata2$x1 == 1 & newdata2$x3 == i], p3a$fit[newdata2$x1 ==
1 & newdata2$x3 == i] + 1.96 * p3a$se.fit[newdata2$x1 == 1 & newdata2$x3 ==
i], col = rgb(0, 0, 1, 0.5), lty = 2)
lines(newdata2$x2[newdata2$x1 == 1 & newdata2$x3 == i], p3a$fit[newdata2$x1 ==
1 & newdata2$x3 == i] - 1.96 * p3a$se.fit[newdata2$x1 == 1 & newdata2$x3 ==
i], col = rgb(0, 0, 1, 0.5), lty = 2)
})
Note how the above code is much more complicated than previously because we now need to draw a separate predicted probability curve (with associated confidence interval) at each level of x3 even though we're not particularly interested in x3. The result is a very confusing plot because the predicted probability curves at each level ofx3 are the essentially the same, but the confidence intervals vary widely because of different levels of certainty due to the sparsity of the original data.
One common response is to simply draw the curve conditional on all other covariates (in this case x3) being at their means, but this is an arbitrary choice. We could also select minimum or maximum, or any other value. Let's write a small function to redraw our curves at different values of x3 to see the impact of this choice:
ppcurve <- function(value_of_x3, title) {
tmp <- expand.grid(x1 = 0:1, x2 = seq(0, 1, length.out = 10), x3 = value_of_x3)
p3tmp <- predict(m3a, tmp, type = "response", se.fit = TRUE)
plot(NA, xlim = c(0, 1), ylim = c(0, 1), xlab = "x2", ylab = "Predicted Probability of y=1",
main = title)
# `x1==0`
lines(tmp$x2[tmp$x1 == 0], p3tmp$fit[tmp$x1 == 0], col = "red")
lines(tmp$x2[tmp$x1 == 0], p3tmp$fit[tmp$x1 == 0] + 1.96 * p3tmp$se.fit[tmp$x1 ==
0], col = "red", lty = 2)
lines(tmp$x2[tmp$x1 == 0], p3tmp$fit[tmp$x1 == 0] - 1.96 * p3tmp$se.fit[tmp$x1 ==
0], col = "red", lty = 2)
# `x1==1`
lines(tmp$x2[tmp$x1 == 1], p3tmp$fit[tmp$x1 == 1], col = "blue")
lines(tmp$x2[tmp$x1 == 1], p3tmp$fit[tmp$x1 == 1] + 1.96 * p3tmp$se.fit[tmp$x1 ==
1], col = "blue", lty = 2)
lines(tmp$x2[tmp$x1 == 1], p3tmp$fit[tmp$x1 == 1] - 1.96 * p3tmp$se.fit[tmp$x1 ==
1], col = "blue", lty = 2)
}
We can then draw a plot that shows the curves for the mean of x3, the minimum of x3 and the maximum of x3.
layout(matrix(1:3, nrow = 1))
ppcurve(mean(x3), title = "x3 at mean")
ppcurve(min(x3), title = "x3 at min")
ppcurve(max(x3), title = "x3 at max")
The above set of plots show that while the inference about the predicted probability curves is the same, the choice of what value of x3 to condition on is meaningful for the confidence intervals. The confidence intervals are much narrower when we condition on the mean value of x3 than the minimum or maximum.
Recall that this model did not properly account for the x1*x2 interaction. Thus while our inference is somewhat sensitive to the choice of conditioning value of the x3 covariate, it is unclear if this minimal sensitivity holds when we properly account for the interaction. Let's take a look at our m3b model that accounts for the interaction.
Let's start by drawing a plot showing the predicted values of the outcome for every combination of x1, x2, and x3:
plot(NA, xlim = c(0, 1), ylim = c(0, 1), xlab = "x2", ylab = "Predicted Probability of y=1")
s <- sapply(unique(newdata2$x3), function(i) {
# `x1==0`
lines(newdata2$x2[newdata2$x1 == 0 & newdata2$x3 == i], p3b$fit[newdata2$x1 ==
0 & newdata2$x3 == i], col = rgb(1, 0, 0, 0.5))
lines(newdata2$x2[newdata2$x1 == 0 & newdata2$x3 == i], p3b$fit[newdata2$x1 ==
0 & newdata2$x3 == i] + 1.96 * p3b$se.fit[newdata2$x1 == 0 & newdata2$x3 ==
i], col = rgb(1, 0, 0, 0.5), lty = 2)
lines(newdata2$x2[newdata2$x1 == 0 & newdata2$x3 == i], p3b$fit[newdata2$x1 ==
0 & newdata2$x3 == i] - 1.96 * p3b$se.fit[newdata2$x1 == 0 & newdata2$x3 ==
i], col = rgb(1, 0, 0, 0.5), lty = 2)
# `x1==1`
lines(newdata2$x2[newdata2$x1 == 1 & newdata2$x3 == i], p3b$fit[newdata2$x1 ==
1 & newdata2$x3 == i], col = rgb(0, 0, 1, 0.5))
lines(newdata2$x2[newdata2$x1 == 1 & newdata2$x3 == i], p3b$fit[newdata2$x1 ==
1 & newdata2$x3 == i] + 1.96 * p3b$se.fit[newdata2$x1 == 1 & newdata2$x3 ==
i], col = rgb(0, 0, 1, 0.5), lty = 2)
lines(newdata2$x2[newdata2$x1 == 1 & newdata2$x3 == i], p3b$fit[newdata2$x1 ==
1 & newdata2$x3 == i] - 1.96 * p3b$se.fit[newdata2$x1 == 1 & newdata2$x3 ==
i], col = rgb(0, 0, 1, 0.5), lty = 2)
})
This plot is incredibly messy. Now, not only our the confidence bands sensitive to what value of x3 we condition on, so too are the predicted probability curves themselves. It is therefore a fairly important decision what level of additional covariates to condition on when estimating the predicted probabilities.
A different approach when deal with interactions is to show marginal effects. Marginal effect, I think, are a bit abstract (i.e., a bit removed from the actual data because they attempt to summarize a lot of information in a single number). The marginal effect is the slope of the curve drawn by taking the difference between, e.g., the predicted probability that y==1 when x1==1 and the predicted probability that y== when x1==0, at each level of x2. Thus, the marginal effect is simply the slope of the difference between the two curves that we were drawing in the above graphs (i.e., the slope of the change in predicted probabilities). Of course, we just saw, if any additional covariate(s) are involved in the data-generating process, then the marginal effect - like the predicted probabilities - is going to differ across levels of that covariate.
Let's see how this works by first returning to our simple interaction model (without x3) and then look at the interaction model with the additional covariate.
To plot the change in predicted probabilities due to x1 across the values of x2, we simply need to take our predicted probabilities from above and difference the values predicted for x1==0 and x1==1. The predicted probabilities for our simple interaction model are stored in p2b, based on new data from newdata1. Let's separate out the values predicted from x1==0 and x1==1 and then take their difference. Let's create a new dataframe that binds newdata1 and the predicted probability and standard error values from p2b together. Then we'll use the split function to that dataframe based upon the value of x1.
tmpdf <- newdata1
tmpdf$fit <- p2b$fit
tmpdf$se.fit <- p2b$se.fit
tmpsplit <- split(tmpdf, tmpdf$x1)
The result is a list of two dataframes, each containing values of x1, x2, and the associated predicted probabilities:
tmpsplit
## $`0`
## x1 x2 fit se.fit
## 1 0 0.0000 0.5014 0.09235
## 3 0 0.1111 0.5011 0.07665
## 5 0 0.2222 0.5007 0.06320
## 7 0 0.3333 0.5003 0.05373
## 9 0 0.4444 0.5000 0.05053
## 11 0 0.5556 0.4996 0.05470
## 13 0 0.6667 0.4992 0.06484
## 15 0 0.7778 0.4989 0.07867
## 17 0 0.8889 0.4985 0.09459
## 19 0 1.0000 0.4982 0.11171
##
## $`1`
## x1 x2 fit se.fit
## 2 1 0.0000 0.3494 0.09498
## 4 1 0.1111 0.3839 0.08187
## 6 1 0.2222 0.4194 0.06887
## 8 1 0.3333 0.4556 0.05769
## 10 1 0.4444 0.4921 0.05089
## 12 1 0.5556 0.5287 0.05093
## 14 1 0.6667 0.5650 0.05770
## 16 1 0.7778 0.6009 0.06862
## 18 1 0.8889 0.6358 0.08115
## 20 1 1.0000 0.6697 0.09362
To calculate the change in predicted probabilty of y==1 due to x1==1 at each value of x2, we'll simply difference the the fit variable from each dataframe:
me <- tmpsplit[[2]]$fit - tmpsplit[[1]]$fit
me
## [1] -0.152032 -0.117131 -0.081283 -0.044766 -0.007877 0.029079 0.065793
## [8] 0.101966 0.137309 0.171555
We also want the standard error of that difference:
me_se <- sqrt(0.5 * (tmpsplit[[2]]$se.fit + tmpsplit[[1]]$se.fit))
Now Let's plot the original predicted probability plot on the left and the change in predicted probability plot on the right:
layout(matrix(1:2, nrow = 1))
plot(NA, xlim = c(0, 1), ylim = c(0, 1), xlab = "x2", ylab = "Predicted Probability of y=1",
main = "Predicted Probabilities")
# `x1==0`
lines(newdata1$x2[newdata1$x1 == 0], p2b$fit[newdata1$x1 == 0], col = "red")
lines(newdata1$x2[newdata1$x1 == 0], p2b$fit[newdata1$x1 == 0] + 1.96 * p2b$se.fit[newdata1$x1 ==
0], col = "red", lty = 2)
lines(newdata1$x2[newdata1$x1 == 0], p2b$fit[newdata1$x1 == 0] - 1.96 * p2b$se.fit[newdata1$x1 ==
0], col = "red", lty = 2)
# `x1==1`
lines(newdata1$x2[newdata1$x1 == 1], p2b$fit[newdata1$x1 == 1], col = "blue")
lines(newdata1$x2[newdata1$x1 == 1], p2b$fit[newdata1$x1 == 1] + 1.96 * p2b$se.fit[newdata1$x1 ==
1], col = "blue", lty = 2)
lines(newdata1$x2[newdata1$x1 == 1], p2b$fit[newdata1$x1 == 1] - 1.96 * p2b$se.fit[newdata1$x1 ==
1], col = "blue", lty = 2)
# plot of change in predicted probabilities:
plot(NA, type = "l", xlim = c(0, 1), ylim = c(-1, 1), xlab = "x2", ylab = "Change in Predicted Probability of y=1",
main = "Change in Predicted Probability due to x1")
abline(h = 0, col = "gray") # gray line at zero
lines(tmpsplit[[1]]$x2, me, lwd = 2) # change in predicted probabilities
lines(tmpsplit[[1]]$x2, me - 1.96 * me_se, lty = 2)
lines(tmpsplit[[1]]$x2, me + 1.96 * me_se, lty = 2)
As should be clear, the plot on the right is simply a further information reduction of the lefthand plot. Where the separate predicted probabilities show the predicted probability of the outcome at each combination of x1 and x2, the righthand plot simply shows the difference between these two curves.
The marginal effect of x2 is thus a further information reduction: it is the slope of the line showing the difference in predicted probabilities.
Because our x2 variable is scaled [0,1], we can see the marginal effect simply by subtracting the value of change in predicted probabilities when x2==0 from the value of change in predicted probabilities when x2==1, which is simply:
me[length(me)] - me[1]
## [1] 0.3236
Thus the marginal effect of x1 on the outcome, is the slope of the line representing the change in predicted probabilities between x1==1 and x1==0 across the range of x2. I don't find that a particularly intuitive measure of effect and would instead prefer to draw some kind of plot rather than reduce that plot to a single number.
Things get more complicated, as we might expect, when we have to account for the additional covariate x3, which influenced our predicted probabilities above. Our predicted probabilies for these data are stored in p3b (based on input data in newdata2). We'll follow the same procedure just used to add those predicted probabilities into a dataframe with the variables from newdata2, then we'll split it based on x1:
tmpdf <- newdata2
tmpdf$fit <- p3b$fit
tmpdf$se.fit <- p3b$se.fit
tmpsplit <- split(tmpdf, tmpdf$x1)
The result is a list of two large dataframes:
str(tmpsplit)
## List of 2
## $ 0:'data.frame': 250 obs. of 5 variables:
## ..$ x1 : int [1:250] 0 0 0 0 0 0 0 0 0 0 ...
## ..$ x2 : num [1:250] 0 0.111 0.222 0.333 0.444 ...
## ..$ x3 : num [1:250] 0 0 0 0 0 0 0 0 0 0 ...
## ..$ fit : num [1:250] 0.437 0.451 0.465 0.479 0.493 ...
## ..$ se.fit: num [1:250] 0.1091 0.0979 0.0893 0.0843 0.0836 ...
## ..- attr(*, "out.attrs")=List of 2
## .. ..$ dim : Named int [1:3] 2 10 25
## .. .. ..- attr(*, "names")= chr [1:3] "x1" "x2" "x3"
## .. ..$ dimnames:List of 3
## .. .. ..$ x1: chr [1:2] "x1=0" "x1=1"
## .. .. ..$ x2: chr [1:10] "x2=0.0000" "x2=0.1111" "x2=0.2222" "x2=0.3333" ...
## .. .. ..$ x3: chr [1:25] "x3=0.0000" "x3=0.2083" "x3=0.4167" "x3=0.6250" ...
## $ 1:'data.frame': 250 obs. of 5 variables:
## ..$ x1 : int [1:250] 1 1 1 1 1 1 1 1 1 1 ...
## ..$ x2 : num [1:250] 0 0.111 0.222 0.333 0.444 ...
## ..$ x3 : num [1:250] 0 0 0 0 0 0 0 0 0 0 ...
## ..$ fit : num [1:250] 0.632 0.642 0.652 0.662 0.672 ...
## ..$ se.fit: num [1:250] 0.1197 0.1047 0.0917 0.0815 0.0749 ...
## ..- attr(*, "out.attrs")=List of 2
## .. ..$ dim : Named int [1:3] 2 10 25
## .. .. ..- attr(*, "names")= chr [1:3] "x1" "x2" "x3"
## .. ..$ dimnames:List of 3
## .. .. ..$ x1: chr [1:2] "x1=0" "x1=1"
## .. .. ..$ x2: chr [1:10] "x2=0.0000" "x2=0.1111" "x2=0.2222" "x2=0.3333" ...
## .. .. ..$ x3: chr [1:25] "x3=0.0000" "x3=0.2083" "x3=0.4167" "x3=0.6250" ...
Now, we need to calculate the change in predicted probability within each of those dataframes, at each value of x3. That is tedious. So let's instead split by both x1 and x3:
tmpsplit <- split(tmpdf, list(tmpdf$x3, tmpdf$x1))
The result is a list of 50 dataframes, the first 25 of which contain data for x1==0 and the latter 25 of which contain data for x1==1:
length(tmpsplit)
## [1] 50
names(tmpsplit)
## [1] "0.0" "0.208333333333333.0" "0.416666666666667.0"
## [4] "0.625.0" "0.833333333333333.0" "1.04166666666667.0"
## [7] "1.25.0" "1.45833333333333.0" "1.66666666666667.0"
## [10] "1.875.0" "2.08333333333333.0" "2.29166666666667.0"
## [13] "2.5.0" "2.70833333333333.0" "2.91666666666667.0"
## [16] "3.125.0" "3.33333333333333.0" "3.54166666666667.0"
## [19] "3.75.0" "3.95833333333333.0" "4.16666666666667.0"
## [22] "4.375.0" "4.58333333333333.0" "4.79166666666667.0"
## [25] "5.0" "0.1" "0.208333333333333.1"
## [28] "0.416666666666667.1" "0.625.1" "0.833333333333333.1"
## [31] "1.04166666666667.1" "1.25.1" "1.45833333333333.1"
## [34] "1.66666666666667.1" "1.875.1" "2.08333333333333.1"
## [37] "2.29166666666667.1" "2.5.1" "2.70833333333333.1"
## [40] "2.91666666666667.1" "3.125.1" "3.33333333333333.1"
## [43] "3.54166666666667.1" "3.75.1" "3.95833333333333.1"
## [46] "4.16666666666667.1" "4.375.1" "4.58333333333333.1"
## [49] "4.79166666666667.1" "5.1"
We can then calculate our change in predicted probabilities at each level of x1 and x3. We'll use the mapply function to do this quickly:
change <- mapply(function(a, b) b$fit - a$fit, tmpsplit[1:25], tmpsplit[26:50])
The resulting object change is a matrix, each column of which is the change in predicted probability at each level of x3. We can then use this matrix to plot each change in predicted probability on a single plot.
Let's again draw this side-by-side with the predicted probability plot:
layout(matrix(1:2, nrow = 1))
# predicted probabilities
plot(NA, xlim = c(0, 1), ylim = c(0, 1), xlab = "x2", ylab = "Predicted Probability of y=1",
main = "Predicted Probabilities")
s <- sapply(unique(newdata2$x3), function(i) {
# `x1==0`
lines(newdata2$x2[newdata2$x1 == 0 & newdata2$x3 == i], p3b$fit[newdata2$x1 ==
0 & newdata2$x3 == i], col = rgb(1, 0, 0, 0.5))
lines(newdata2$x2[newdata2$x1 == 0 & newdata2$x3 == i], p3b$fit[newdata2$x1 ==
0 & newdata2$x3 == i] + 1.96 * p3b$se.fit[newdata2$x1 == 0 & newdata2$x3 ==
i], col = rgb(1, 0, 0, 0.5), lty = 2)
lines(newdata2$x2[newdata2$x1 == 0 & newdata2$x3 == i], p3b$fit[newdata2$x1 ==
0 & newdata2$x3 == i] - 1.96 * p3b$se.fit[newdata2$x1 == 0 & newdata2$x3 ==
i], col = rgb(1, 0, 0, 0.5), lty = 2)
# `x1==1`
lines(newdata2$x2[newdata2$x1 == 1 & newdata2$x3 == i], p3b$fit[newdata2$x1 ==
1 & newdata2$x3 == i], col = rgb(0, 0, 1, 0.5))
lines(newdata2$x2[newdata2$x1 == 1 & newdata2$x3 == i], p3b$fit[newdata2$x1 ==
1 & newdata2$x3 == i] + 1.96 * p3b$se.fit[newdata2$x1 == 1 & newdata2$x3 ==
i], col = rgb(0, 0, 1, 0.5), lty = 2)
lines(newdata2$x2[newdata2$x1 == 1 & newdata2$x3 == i], p3b$fit[newdata2$x1 ==
1 & newdata2$x3 == i] - 1.96 * p3b$se.fit[newdata2$x1 == 1 & newdata2$x3 ==
i], col = rgb(0, 0, 1, 0.5), lty = 2)
})
# change in predicted probabilities
plot(NA, type = "l", xlim = c(0, 1), ylim = c(-1, 1), xlab = "x2", ylab = "Change in Predicted Probability of y=1",
main = "Change in Predicted Probability due to x1")
abline(h = 0, col = "gray")
apply(change, 2, function(a) lines(tmpsplit[[1]]$x2, a))
## NULL
As we can see, despite the craziness of the left-hand plot, the marginal effect of x1 is actually not affected by x3 (which makes sense because it is not interacted with x3 in the data-generating process). Thus, while the choice of value of x3 on which to estimated the predicted probabilities matters, the marginal effect is constant. We can estimate it simply by following the same procedure above from any column of our change matrix:
change[nrow(change), 1] - change[1, 1]
## 0.0
## -0.0409
The result here is a negligible marginal effect, which is what we would expect given the lack of an interaction between x1 and x3 in the underlying data. If such an interaction were in the actual data, then we should expect that this marginal effect would vary across values of x3 and we would need to further state the marginal effect as conditional on a particular value of x3.