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A typical use of a logarithmic transformation variable is to pull outlying data from a positively skewed distribution closer to the bulk of the data in a quest to have the variable be normally distributed. In regression analysis the logs of variables are routinely taken, not necessarily for achieving a normal distribution of the predictors and/or the dependent variable but for interpretability. Standard interpretation of coefficients in a regression analysis is that a one unit change in the independent variable results in the respective regression coefficient (beta) change in the expected value of the dependent variable while all the predictors are held constant. Interpreting a log transformed variable can be done in such a manner, however, such coefficients are routinely interpreted in terms of percent change (See Introductory Econometrics: A Modern Approach by Woolridge for discussion and derivation). Throughout this page we'll explore the interpretation in a simple linear regression setting with either the dependent variable, independent variable, or both variables are log-transformed. We'll use the hospital-level data from the Study on the Efficacy of Nosocomial Infection Control (data came from Applied Linear Regression Models 5th edition) where we'll explore the relationship between average length of stay (in days) for all patients in the hospital (length) and the average daily number of patients in the hospital (census). The focus of this page is model interpretation, not model logistics.
NOTE: The ensuing interpretation is applicable for only log base e (natural log) transformations.
data list list/id length age risk culture xray beds msch region census nurses svcs. begin data. 1 7.13 55.7 4.1 9.0 39.6 279 2 4 207 241 60.0 2 8.82 58.2 1.6 3.8 51.7 80 2 2 51 52 40.0 3 8.34 56.9 2.7 8.1 74.0 107 2 3 82 54 20.0 4 8.95 53.7 5.6 18.9 122.8 147 2 4 53 148 40.0 5 11.20 56.5 5.7 34.5 88.9 180 2 1 134 151 40.0 6 9.76 50.9 5.1 21.9 97.0 150 2 2 147 106 40.0 7 9.68 57.8 4.6 16.7 79.0 186 2 3 151 129 40.0 8 11.18 45.7 5.4 60.5 85.8 640 1 2 399 360 60.0 9 8.67 48.2 4.3 24.4 90.8 182 2 3 130 118 40.0 10 8.84 56.3 6.3 29.6 82.6 85 2 1 59 66 40.0 11 11.07 53.2 4.9 28.5 122.0 768 1 1 591 656 80.0 12 8.30 57.2 4.3 6.8 83.8 167 2 3 105 59 40.0 13 12.78 56.8 7.7 46.0 116.9 322 1 1 252 349 57.1 14 7.58 56.7 3.7 20.8 88.0 97 2 2 59 79 37.1 15 9.00 56.3 4.2 14.6 76.4 72 2 3 61 38 17.1 16 11.08 50.2 5.5 18.6 63.6 387 2 3 326 405 57.1 17 8.28 48.1 4.5 26.0 101.8 108 2 4 84 73 37.1 18 11.62 53.9 6.4 25.5 99.2 133 2 1 113 101 37.1 19 9.06 52.8 4.2 6.9 75.9 134 2 2 103 125 37.1 20 9.35 53.8 4.1 15.9 80.9 833 2 3 547 519 77.1 21 7.53 42.0 4.2 23.1 98.9 95 2 4 47 49 17.1 22 10.24 49.0 4.8 36.3 112.6 195 2 2 163 170 37.1 23 9.78 52.3 5.0 17.6 95.9 270 1 1 240 198 57.1 24 9.84 62.2 4.8 12.0 82.3 600 2 3 468 497 57.1 25 9.20 52.2 4.0 17.5 71.1 298 1 4 244 236 57.1 26 8.28 49.5 3.9 12.0 113.1 546 1 2 413 436 57.1 27 9.31 47.2 4.5 30.2 101.3 170 2 1 124 173 37.1 28 8.19 52.1 3.2 10.8 59.2 176 2 1 156 88 37.1 29 11.65 54.5 4.4 18.6 96.1 248 2 1 217 189 37.1 30 9.89 50.5 4.9 17.7 103.6 167 2 2 113 106 37.1 31 11.03 49.9 5.0 19.7 102.1 318 2 1 270 335 57.1 32 9.84 53.0 5.2 17.7 72.6 210 2 2 200 239 54.3 33 11.77 54.1 5.3 17.3 56.0 196 2 1 164 165 34.3 34 13.59 54.0 6.1 24.2 111.7 312 2 1 258 169 54.3 35 9.74 54.4 6.3 11.4 76.1 221 2 2 170 172 54.3 36 10.33 55.8 5.0 21.2 104.3 266 2 1 181 149 54.3 37 9.97 58.2 2.8 16.5 76.5 90 2 2 69 42 34.3 38 7.84 49.1 4.6 7.1 87.9 60 2 3 50 45 34.3 39 10.47 53.2 4.1 5.7 69.1 196 2 2 168 153 54.3 40 8.16 60.9 1.3 1.9 58.0 73 2 3 49 21 14.3 41 8.48 51.1 3.7 12.1 92.8 166 2 3 145 118 34.3 42 10.72 53.8 4.7 23.2 94.1 113 2 3 90 107 34.3 43 11.20 45.0 3.0 7.0 78.9 130 2 3 95 56 34.3 44 10.12 51.7 5.6 14.9 79.1 362 1 3 313 264 54.3 45 8.37 50.7 5.5 15.1 84.8 115 2 2 96 88 34.3 46 10.16 54.2 4.6 8.4 51.5 831 1 4 581 629 74.3 47 19.56 59.9 6.5 17.2 113.7 306 2 1 273 172 51.4 48 10.90 57.2 5.5 10.6 71.9 593 2 2 446 211 51.4 49 7.67 51.7 1.8 2.5 40.4 106 2 3 93 35 11.4 50 8.88 51.5 4.2 10.1 86.9 305 2 3 238 197 51.4 51 11.48 57.6 5.6 20.3 82.0 252 2 1 207 251 51.4 52 9.23 51.6 4.3 11.6 42.6 620 2 2 413 420 71.4 53 11.41 61.1 7.6 16.6 97.9 535 2 3 330 273 51.4 54 12.07 43.7 7.8 52.4 105.3 157 2 2 115 76 31.4 55 8.63 54.0 3.1 8.4 56.2 76 2 1 39 44 31.4 56 11.15 56.5 3.9 7.7 73.9 281 2 1 217 199 51.4 57 7.14 59.0 3.7 2.6 75.8 70 2 4 37 35 31.4 58 7.65 47.1 4.3 16.4 65.7 318 2 4 265 314 51.4 59 10.73 50.6 3.9 19.3 101.0 445 1 2 374 345 51.4 60 11.46 56.9 4.5 15.6 97.7 191 2 3 153 132 31.4 61 10.42 58.0 3.4 8.0 59.0 119 2 1 67 64 31.4 62 11.18 51.0 5.7 18.8 55.9 595 1 2 546 392 68.6 63 7.93 64.1 5.4 7.5 98.1 68 2 4 42 49 28.6 64 9.66 52.1 4.4 9.9 98.3 83 2 2 66 95 28.6 65 7.78 45.5 5.0 20.9 71.6 489 2 3 391 329 48.6 66 9.42 50.6 4.3 24.8 62.8 508 2 1 421 528 48.6 67 10.02 49.5 4.4 8.3 93.0 265 2 2 191 202 48.6 68 8.58 55.0 3.7 7.4 95.9 304 2 3 248 218 48.6 69 9.61 52.4 4.5 6.9 87.2 487 2 3 404 220 48.6 70 8.03 54.2 3.5 24.3 87.3 97 2 1 65 55 28.6 71 7.39 51.0 4.2 14.6 88.4 72 2 2 38 67 28.6 72 7.08 52.0 2.0 12.3 56.4 87 2 3 52 57 28.6 73 9.53 51.5 5.2 15.0 65.7 298 2 3 241 193 48.6 74 10.05 52.0 4.5 36.7 87.5 184 1 1 144 151 68.6 75 8.45 38.8 3.4 12.9 85.0 235 2 2 143 124 48.6 76 6.70 48.6 4.5 13.0 80.8 76 2 4 51 79 28.6 77 8.90 49.7 2.9 12.7 86.9 52 2 1 37 35 28.6 78 10.23 53.2 4.9 9.9 77.9 752 1 2 595 446 68.6 79 8.88 55.8 4.4 14.1 76.8 237 2 2 165 182 48.6 80 10.30 59.6 5.1 27.8 88.9 175 2 2 113 73 45.7 81 10.79 44.2 2.9 2.6 56.6 461 1 2 320 196 65.7 82 7.94 49.5 3.5 6.2 92.3 195 2 2 139 116 45.7 83 7.63 52.1 5.5 11.6 61.1 197 2 4 109 110 45.7 84 8.77 54.5 4.7 5.2 47.0 143 2 4 85 87 25.7 85 8.09 56.9 1.7 7.6 56.9 92 2 3 61 61 45.7 86 9.05 51.2 4.1 20.5 79.8 195 2 3 127 112 45.7 87 7.91 52.8 2.9 11.9 79.5 477 2 3 349 188 65.7 88 10.39 54.6 4.3 14.0 88.3 353 2 2 223 200 65.7 89 9.36 54.1 4.8 18.3 90.6 165 2 1 127 158 45.7 90 11.41 50.4 5.8 23.8 73.0 424 1 3 359 335 45.7 91 8.86 51.3 2.9 9.5 87.5 100 2 3 65 53 25.7 92 8.93 56.0 2.0 6.2 72.5 95 2 3 59 56 25.7 93 8.92 53.9 1.3 2.2 79.5 56 2 2 40 14 5.7 94 8.15 54.9 5.3 12.3 79.8 99 2 4 55 71 25.7 95 9.77 50.2 5.3 15.7 89.7 154 2 2 123 148 25.7 96 8.54 56.1 2.5 27.0 82.5 98 2 1 57 75 45.7 97 8.66 52.8 3.8 6.8 69.5 246 2 3 178 177 45.7 98 12.01 52.8 4.8 10.8 96.9 298 2 1 237 115 45.7 99 7.95 51.8 2.3 4.6 54.9 163 2 3 128 93 42.9 100 10.15 51.9 6.2 16.4 59.2 568 1 3 452 371 62.9 101 9.76 53.2 2.6 6.9 80.1 64 2 4 47 55 22.9 102 9.89 45.2 4.3 11.8 108.7 190 2 1 141 112 42.9 103 7.14 57.6 2.7 13.1 92.6 92 2 4 40 50 22.9 104 13.95 65.9 6.6 15.6 133.5 356 2 1 308 182 62.9 105 9.44 52.5 4.5 10.9 58.5 297 2 3 230 263 42.9 106 10.80 63.9 2.9 1.6 57.4 130 2 3 69 62 22.9 107 7.14 51.7 1.4 4.1 45.7 115 2 3 90 19 22.9 108 8.02 55.0 2.1 3.8 46.5 91 2 2 44 32 22.9 109 11.80 53.8 5.7 9.1 116.9 571 1 2 441 469 62.9 110 9.50 49.3 5.8 42.0 70.9 98 2 3 68 46 22.9 111 7.70 56.9 4.4 12.2 67.9 129 2 4 85 136 62.9 112 17.94 56.2 5.9 26.4 91.8 835 1 1 791 407 62.9 113 9.41 59.5 3.1 20.6 91.7 29 2 3 20 22 22.9 end data.
We'll start of by looking at histograms of the length and census variable in its original metric and log-transformed state.
compute loglength = ln(length). compute logcensus = ln(census). exe. graph /histogram = length.graph /histogram = loglength.
graph /histogram = census.
graph /histogram = logcensus.
In both graphs, we saw how taking a log-transformation of the variable brought the outlying data points from the right tail towards the rest of the data. We'll start off by interpreting a linear regression model were the variables are in their original metric and then proceed to include the variables in their transformed state.
For the first model with the variables in their original state, we'll regress average length of stay on the average daily number of patients in the hospital. Interpretation of the relationship is that a one person increase in the average daily number of patients in the hospital will change the average length of stay by 0.006 days.
regression /dependent length /method enter = census.
In this model, the dependent variable is in its log-transformed state and the independent variable is in its original metric. Comparing beta coefficient for census to the prior model we note that there is a big difference in coefficients, however we must recall the scale of the dependent variable changed states. In such models where the dependent variable has been log-transformed and the predictors are not, the format for interpretation is that dependent variable changes by 100*(beta coefficient) percent for a one unit increase in the independent variable while all other variable in the model are held constant. In this particular model we'd say that an increase of one person in the average daily number of patients in the hospital would results in a 0.055 percent change in the average length of stay (the beta coefficient for the independent variable was rounded to the nearest thousandths, the more precise beta coefficient is 0.00055).
regression /dependent loglength /method enter = census.
In this model we are going to switch by having the dependent variable in its original metric and the independent variable being log-transformed. A comparison to the prior two models reveals that the regression coefficient is drastically different. Similar to the prior example the interpretation has a nice format, a one percent increase in the independent variable increases (or decreases) the dependent variable by (beta/100) units. In this particular model we'd say that a one percent increase in the average daily number of patients in the hospital would result in a (1.155/100) = 0.016 day increase in the average length of stay.
regression /dependent length /method enter = logcensus.
Instances where both the dependent variables and independent variable(s) are log-transformed variables, the relationship is commonly referred in econometrics as elasticity. In a regression setting, we'd interpret the elasticity as the percent change in y (the dependent variable), while x (the independent variable) increases by one percent. For this model we'd conclude that a one percent increase in the average daily number of patients in the hospital would yield a 0.11% increase in the average length of stay.
regression /dependent loglength /method enter = logcensus.
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