Medical costs represent a significant financial burden for individuals and are a primary reason health insurance coverage is widely sought. Accurately understanding the factors that drive medical insurance charges is essential for insurers seeking to price plans fairly and sustainably, as well as for policymakers concerned with healthcare affordability. This paper aims to model the relationship between individual medical insurance costs and selected demographic, health-related, and behavioral indicators using linear regression techniques. Using a dataset of insured individuals in the United States, we examine how factors such as age, body mass index (BMI), smoking status, number of dependents, and geographic region influence medical charges. Multiple regression models are developed and evaluated to identify statistically significant predictors and assess model assumptions. The results provide interpretable insights into the primary drivers of medical insurance costs and demonstrate the usefulness of regression modeling in health economics and insurance pricing applications.
Abstract
Medical insurance costs are highly variable, right-skewed, and influenced by both demographic and behavioral factors. Accurately modeling these costs is essential for insurers seeking to price risk fairly and sustainably. This study examines the relationship between medical insurance charges and individual characteristics using regression-based approaches. Using a dataset of 1,338 insured individuals in the United States, we evaluate ordinary least squares regression, interaction models, robust regression, and generalized linear models. Due to the skewed and heteroskedastic nature of cost data, a Gamma generalized linear model with a log link and selected interaction terms was chosen as the final specification. Results indicate that smoking status is the dominant driver of medical costs and significantly amplifies the effect of body mass index. The final model demonstrates strong predictive performance on a held-out test set and provides interpretable insights relevant for insurance pricing and risk assessment.
1. Introduction
Rising healthcare costs present a significant financial challenge for individuals, insurers, and policymakers. Medical insurance premiums are typically based on expected healthcare utilization, which in turn depends on demographic characteristics, health indicators, and behavioral risk factors. Understanding how these factors jointly influence medical costs is critical for effective risk pricing and sustainable insurance design.
Traditional linear regression models are often used to analyze cost data due to their interpretability; however, medical costs violate many classical regression assumptions. Costs are strictly positive, right-skewed, and characterized by a small number of high-cost individuals who account for a disproportionate share of total expenditures. As a result, naïve linear regression may produce biased inference and poor predictive performance.
This study addresses these challenges by applying regression techniques tailored to cost data. Specifically, we investigate whether smoking status modifies the effects of other predictors—particularly body mass index—on medical insurance charges. The objective is twofold: (1) to identify key drivers of medical costs and (2) to develop a predictive model that balances interpretability with statistical appropriateness.
2. Literature Review
Modeling healthcare expenditures presents significant econometric challenges due to the inherently skewed and heavy-tailed nature of cost data. Medical expenditures are characterized by a large mass of low-cost observations and a long right tail of extremely high costs. These features are commonly associated with heteroscedastic error structures, leading to frequent violations of the homoscedasticity assumption in ordinary least squares regression.
A common response is to apply a logarithmic transformation to costs and estimate a linear model on the transformed scale. However, this approach introduces important limitations. Zero-cost observations cannot be accommodated without ad hoc adjustments, and retransformation bias arises when predicted log-costs are exponentiated to recover expected costs on the original scale. As shown by Manning (1998), this bias persists under heteroscedasticity, which is common even after log transformation. While smearing estimators may partially address this issue (Duan, 1983; Duan et al., 1983), they rely on additional assumptions and do not fully resolve model misspecification (Manning, 1998).
Generalized linear models provide a more appropriate framework for modeling healthcare costs (Blough et al., 1999; Deb & Trivedi, 2002). By allowing the response variable to follow a distribution from the exponential dispersion family, GLMs explicitly accommodate the skewed and heteroscedastic nature of cost data (Glick et al., 2014). In particular, the Gamma distribution with a log link is well suited for continuous, positive expenditures, as it naturally models variance proportional to the square of the mean and ensures positive predictions (Blough et al., 1999; Deb & Trivedi, 2002). Coefficients from such models admit a straightforward multiplicative interpretation in terms of percentage changes in expected cost (Glick et al., 2014).
Consistent with prior literature, our analysis highlights smoking status as a dominant cost driver and demonstrates that its effect interacts strongly with body mass index. This finding aligns with evidence that health risk factors operate synergistically rather than independently (Manning et al., 1987; Deb & Trivedi, 2002). Age and family structure also exhibit significant relationships with medical expenditures, reflecting systematic differences in healthcare utilization documented in prior studies (Blough et al., 1999; Glick et al., 2014).
Overall, the use of a Gamma generalized linear model with targeted interaction terms aligns with best practices in health econometrics, yielding consistent estimates and improved efficiency relative to misspecified linear models (Manning, 1998; Blough et al., 1999; Deb & Trivedi, 2002).
3. Methodology
3.1 Data
The dataset is found from the online Data Science community Kaggle.com. A link to the exact download source is provided in the Appendix.
The dataset consists of 1,338 individuals covered by medical insurance in the United States. The response variable is annual medical insurance charges. Predictor variables include age, sex, body mass index (BMI), smoking status, number of dependents (recoded as a binary indicator for having children), and residential region.
All variables were complete with no missing observations. Categorical predictors were encoded as factor variables prior to model estimation. The number of children was originally measured as a numeric variable and was subsequently recoded as a binary indicator reflecting the presence of dependents.
3.2 Exploratory Analysis
Exploratory data analysis revealed that medical charges are heavily right-skewed, with a long upper tail corresponding to high-cost individuals. Boxplots and residual diagnostics confirmed the presence of heteroskedasticity and non-normality, consistent with the known characteristics of healthcare cost data. These patterns motivated the use of modeling approaches beyond ordinary least squares regression.
3.3 Model Specification
Several models were estimated and compared:
Ordinary least squares regression on raw charges
Linear regression with interaction terms
Robust regression to assess sensitivity to outliers
Gamma generalized linear models with a log link
To assess effect heterogeneity, interaction terms between smoking status and other predictors were explored. While smoking significantly modified the effects of BMI, age, and dependent status, interactions with sex and region did not materially improve model fit.
The final model specification was a Gamma generalized linear model with a log link including the following predictors: Age, BMI, Smoking status,Having children, Sex, Region, Smoking × BMI, Smoking, Age, Smoking × Having children
3.4 Model Evaluation
The dataset was split into training (70%) and testing (30%) subsets. Model performance was evaluated on the test set using root mean squared error (RMSE) and mean absolute error (MAE), with predictions generated on the original cost scale using the inverse log link.
4. Results
The final Gamma model demonstrated strong explanatory power, reducing deviance substantially relative to a null model and achieving a deviance-based pseudo-R² of approximately 0.73.
Smoking status emerged as the dominant predictor of medical costs. Holding other factors constant, smokers incur nearly three times the expected medical costs of non-smokers. Body mass index alone had little effect among non-smokers; however, for smokers, each additional BMI unit increased expected medical costs by approximately 5%. This interaction indicates that smoking substantially amplifies the cost impact of obesity.
Age was also a significant predictor, with expected medical costs increasing by roughly 3–4% per additional year for non-smokers. The interaction between smoking and age suggested that the marginal effect of age is attenuated among smokers, likely reflecting their already elevated baseline risk. Having dependents increased expected costs for non-smokers, though this effect was weaker among smokers.
On the held-out test set, the model achieved an RMSE of approximately $5,200 and an MAE of approximately $3,170. A predicted-versus-actual plot showed good calibration across most of the cost distribution, with increased dispersion among very high-cost individuals.
5. Discussion
The results highlight smoking as the most influential driver of medical insurance costs, both directly and through its interaction with BMI and age. From a business perspective, these findings support risk-adjusted pricing strategies that account not only for smoking status but also for how smoking interacts with other health indicators.
The strong interaction between smoking and BMI suggests that wellness interventions targeting weight management among smokers may yield disproportionately large cost savings. Insurers may also benefit from differentiated premium structures or targeted preventive care programs for high-risk subgroups identified by the model.
6. Limitations
Several limitations should be noted. The dataset does not include detailed clinical variables such as diagnosed conditions or healthcare utilization patterns, which likely explain much of the remaining variation in high-cost cases. Additionally, the analysis focuses on expected costs rather than extreme tail risk, which may be of interest for reinsurance or catastrophic coverage modeling.
7. Conclusion
This study demonstrates that medical insurance costs can be effectively modeled using regression-based approaches when the distributional characteristics of cost data are properly addressed. A Gamma generalized linear model with targeted interaction terms provides a robust, interpretable framework for understanding cost drivers and predicting expected medical expenditures. The findings underscore the central role of smoking in healthcare costs and illustrate the value of interaction modeling in uncovering meaningful risk heterogeneity.
Medical insurance charges as a function of body mass index (BMI), stratified by smoking status. Points represent observed charges, while solid lines represent fitted values from the final Gamma generalized linear model with a log link and interaction between BMI and smoking status. Predicted values are shown on the original cost scale.
The mdoel's rmse: 5198.975 The models mae 3170.379
Appendices
Models
Call:
lm(formula = charges ~ ., data = data_clean)
Residuals:
Min 1Q Median 3Q Max
-11483.5 -2894.7 -956.5 1478.1 30059.6
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -12001.01 993.99 -12.074 < 2e-16 ***
age 256.91 11.92 21.562 < 2e-16 ***
sexmale -126.41 333.31 -0.379 0.70456
bmi 339.51 28.63 11.858 < 2e-16 ***
smokeryes 23849.65 413.63 57.660 < 2e-16 ***
regionnorthwest -352.22 476.88 -0.739 0.46029
regionsoutheast -1057.33 479.27 -2.206 0.02755 *
regionsouthwest -944.26 478.40 -1.974 0.04861 *
has_childrenYes 999.58 335.88 2.976 0.00297 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 6069 on 1329 degrees of freedom
Multiple R-squared: 0.7503, Adjusted R-squared: 0.7488
F-statistic: 499.3 on 8 and 1329 DF, p-value: < 2.2e-16
Call:
lm(formula = charges ~ bmi * smoker + age + has_children, data = data_clean)
Residuals:
Min 1Q Median 3Q Max
-14992.4 -2033.6 -1242.8 -372.9 29880.8
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -2753.520 838.589 -3.284 0.001052 **
bmi 6.423 24.950 0.257 0.796894
smokeryes -20082.183 1659.601 -12.101 < 2e-16 ***
age 265.192 9.585 27.667 < 2e-16 ***
has_childrenYes 960.640 270.224 3.555 0.000391 ***
bmi:smokeryes 1430.143 52.987 26.991 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 4886 on 1332 degrees of freedom
Multiple R-squared: 0.8378, Adjusted R-squared: 0.8372
F-statistic: 1376 on 5 and 1332 DF, p-value: < 2.2e-16
Call:
lm(formula = sqrt(charges) ~ (bmi * smoker + age), data = data_clean)
Residuals:
Min 1Q Median 3Q Max
-32.971 -9.796 -4.530 0.831 107.572
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 28.14841 3.49594 8.052 1.79e-15 ***
bmi 0.05813 0.10529 0.552 0.581
smokeryes -31.44669 7.00376 -4.490 7.74e-06 ***
age 1.43666 0.04041 35.554 < 2e-16 ***
bmi:smokeryes 3.97785 0.22361 17.789 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 20.62 on 1333 degrees of freedom
Multiple R-squared: 0.8142, Adjusted R-squared: 0.8137
F-statistic: 1461 on 4 and 1333 DF, p-value: < 2.2e-16
Selected Model
Call:
glm(formula = charges ~ age + bmi + has_children + sex + region +
smoker + smoker:bmi + smoker:age + smoker:has_children, family = Gamma(link = "log"),
data = data_clean)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 7.467355 0.127288 58.665 < 2e-16 ***
age 0.034462 0.001534 22.466 < 2e-16 ***
bmi 0.002258 0.003690 0.612 0.54075
has_childrenYes 0.270674 0.043227 6.262 5.13e-10 ***
sexmale -0.063918 0.038358 -1.666 0.09588 .
regionnorthwest -0.066977 0.054828 -1.222 0.22208
regionsoutheast -0.151504 0.055130 -2.748 0.00608 **
regionsouthwest -0.169059 0.055015 -3.073 0.00216 **
smokeryes 1.080839 0.266440 4.057 5.27e-05 ***
bmi:smokeryes 0.049428 0.007595 6.508 1.08e-10 ***
age:smokeryes -0.025914 0.003422 -7.573 6.78e-14 ***
has_childrenYes:smokeryes -0.274594 0.096193 -2.855 0.00438 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for Gamma family taken to be 0.4861963)
Null deviance: 1056.04 on 1337 degrees of freedom
Residual deviance: 288.82 on 1326 degrees of freedom
AIC: 26168
Number of Fisher Scoring iterations: 6
References
Manning, W. G. (1998). The logged dependent variable, heteroscedasticity, and the retransformation problem. Journal of Health Economics, 17(3), 283–295. https://doi.org/10.1016/S0167-6296(98)00025-3
Blough, D. K., Madden, C. W., & Hornbrook, M. C. (1999). Modeling risk using generalized linear models. Journal of Health Economics, 18(2), 153–171. https://doi.org/10.1016/S0167-6296(98)00041-1
Deb, P., & Trivedi, P. K. (2002). The structure of demand for health care by the elderly: Econometric evidence from the Health and Retirement Study. Journal of Health Economics, 21(5), 803–824. https://doi.org/10.1016/S0167-6296(02)00021-7
Glick, H. A., Doshi, J. A., Sonnad, S. S., & Polsky, D. (2014). Economic evaluation in clinical trials (2nd ed.). Oxford University Press.
Predictions
Predicted charges
1 21667.670
2 5038.960
3 4639.096
4 4122.332
5 29303.890
6 3309.855
7 15008.039
8 35764.482
9 15374.077
10 3453.264
11 43254.124
12 7563.904
13 4722.729
14 18232.641
15 18171.631
16 20064.448
17 14671.339
18 5725.687
19 3152.160
20 7242.559
21 8996.259
22 39299.724
23 8503.573
24 11575.900
25 10236.835
26 15330.048
27 3975.315
28 4082.238
29 9001.299
30 3220.740
31 3480.834
32 35483.043
33 3130.432
34 3720.357
35 3419.793
36 20478.467
37 5846.825
38 25659.763
39 5846.385
40 32297.263
41 3729.171
42 24713.371
43 33711.791
44 11759.760
45 34723.498
46 12776.781
47 11585.490
48 6965.176
49 4436.735
50 13583.932
51 8033.744
52 10091.537
53 12085.888
54 3049.048
55 4804.498
56 57151.096
57 5421.165
58 3538.956
59 3166.220
60 17356.019
61 13921.274
62 9793.813
63 5167.157
64 3623.992
65 11718.099
66 29035.963
67 4035.101
68 16036.034
69 2960.915
70 3498.093
71 6241.120
72 16166.321
73 41529.979
74 6099.973
75 27709.145
76 26419.087
77 54772.624
78 9172.891
79 3866.868
80 6440.580
81 9584.883
82 23015.811
83 34746.569
84 7918.280
85 5574.483
86 5894.672
87 26843.127
88 56742.537
89 13599.835
90 18141.460
91 3596.394
92 3146.587
93 4423.483
94 7200.641
95 9965.981
96 6503.324
97 4239.782
98 13308.270
99 14046.840
100 13119.523
101 25362.868
102 25954.463
103 10115.530
104 7262.874
105 2883.653
106 12122.143
107 12042.383
108 9214.050
109 3120.227
110 8976.342
111 16000.059
112 15013.187
113 7185.584
114 21295.282
115 15603.567
116 3410.133
117 18334.933
118 4913.227
119 6100.487
120 12517.132
121 3763.969
122 6479.874
123 13148.691
124 32620.192
125 3341.873
126 11522.443
127 27578.707
128 11990.163
129 6360.497
130 3870.823
131 3851.262
132 9531.281
133 10893.413
134 46164.010
135 3483.270
136 7662.204
137 7796.128
138 4236.471
139 3859.879
140 11089.481
141 10427.555
142 10954.087
143 11479.932
144 14028.825
145 4219.745
146 10568.079
147 26050.287
148 37767.908
149 3201.730
150 11266.741
151 6110.268
152 17672.584
153 47964.989
154 17410.496
155 6264.738
156 2795.826
157 7827.168
158 10625.523
159 6347.110
160 8959.510
161 14013.510
162 19540.276
163 27123.798
164 7762.654
165 3507.005
166 23232.261
167 5485.110
168 11000.312
169 31933.766
170 12515.015
171 6346.780
172 52600.604
173 12246.816
174 11524.444
175 16239.518
176 4364.064
177 26440.784
178 13745.311
179 14503.806
180 5128.654
181 8814.707
182 13391.784
183 8436.238
184 12438.585
185 10493.044
186 36798.494
187 43000.292
188 5584.397
189 5100.489
190 6626.366
191 62153.607
192 13580.717
193 6297.897
194 6151.934
195 3600.765
196 9961.443
197 4759.045
198 10844.973
199 9624.108
200 11584.839
201 6506.011
202 11646.411
203 10119.098
204 3556.025
205 11699.757
206 2890.171
207 11847.040
208 12755.827
209 12558.742
210 44230.821
211 10712.304
212 23628.389
213 3103.690
214 5308.774
215 4346.560
216 21896.021
217 9874.117
218 8794.058
219 14668.694
220 16134.489
221 8930.916
222 10179.478
223 2878.786
224 6129.669
225 3440.502
226 2839.720
227 3935.700
228 8840.701
229 24862.980
230 23107.288
231 3940.617
232 8366.667
233 9349.060
234 3421.122
235 5399.116
236 12468.616
237 5942.402
238 6911.615
239 4365.245
240 29623.213
241 40122.446
242 9245.661
243 9605.961
244 12823.894
245 8106.882
246 13097.682
247 5998.925
248 7510.882
249 8528.811
250 12850.292
251 8080.127
252 3066.482
253 29249.870
254 33680.362
255 4577.770
256 11002.150
257 17715.896
258 26784.153
259 3147.475
260 69684.126
261 8512.293
262 2853.872
263 14300.164
264 4843.401
265 8560.028
266 7410.493
267 4195.532
268 5059.101
269 24285.941
270 7278.700
271 11143.693
272 4103.520
273 3062.709
274 5606.430
275 12647.804
276 9660.987
277 20417.253
278 4578.860
279 17405.785
280 3713.274
281 12888.910
282 7114.699
283 6429.207
284 12960.834
285 14453.080
286 8207.225
287 5032.439
288 6059.125
289 12011.489
290 2764.579
291 12254.692
292 58812.648
293 4888.774
294 4256.307
295 13123.284
296 9708.068
297 8168.276
298 11768.525
299 5956.519
300 3743.975
301 5560.207
302 20230.398
303 4588.617
304 18286.786
305 11653.883
306 9270.388
307 8022.143
308 3360.030
309 28953.375
310 13283.566
311 17394.932
312 11318.753
313 4762.498
314 8738.167
315 39951.595
316 13701.631
317 34136.002
318 4119.393
319 27691.623
320 28035.041
321 4111.198
322 15507.412
323 30673.409
324 4356.023
325 5575.594
326 18529.496
327 15452.301
328 3589.969
329 17495.247
330 3612.261
331 11754.378
332 12362.150
333 4578.455
334 48394.852
335 5038.204
336 18275.063
337 10376.614
338 12750.563
339 9130.038
340 13250.046
341 28716.735
342 6258.094
343 56979.174
344 7141.252
345 2966.267
346 12819.080
347 10237.145
348 30421.248
349 11264.167
350 8565.498
351 12695.240
352 13115.022
353 11947.389
354 8338.360
355 6564.420
356 8292.702
357 23117.378
358 7964.396
359 5878.260
360 3353.081
361 12758.564
362 23432.951
363 33713.098
364 14924.821
365 27389.933
366 6723.463
367 7609.264
368 6999.639
369 16006.883
370 12533.446
371 38435.757
372 8508.315
373 6777.903
374 9330.905
375 6725.493
376 10504.062
377 12973.073
378 15944.054
379 14581.824
380 16112.323
381 12872.568
382 3059.668
383 27756.090
384 2790.151
385 6844.843
386 9120.913
387 10535.917
388 29585.583
389 10427.464
390 5272.620
391 29968.411
392 3238.056
393 37765.224
394 4805.280
395 24825.793
396 8027.887
397 6461.406
398 30672.450
399 6210.509
400 3520.365
401 3039.221
402 3230.003
