Model for Generic Proportion: Linear Logit Model

Model for generic proportion: linear logit model

Formula

Logit (generic proportioni) = β1+ β2*timei+ β3*monthi² + β4*monthi³ + βcp1*I{timei>cp1}

+ βcp1*(timei-cp1)*I{timei>cp1} + … +βcpk*I{timei>cpk} + βcpk*(timei-cpk)*I{timei>cpk} + εi

andεi~ N(0, σ²)

Goodness of fit statistics

Statistic / NIHDI costs / Co-payment costs / Packages sold / DDD sold / Units sold
Acid blocking agents / -2 Log Likelihood / -53.80 / -54.99 / -94.55 / -104.16 / -101.89
AIC / -21.80 / -20.99 / -60.55 / -70.16 / -67.89
BIC / 21.84 / 25.23 / -14.34 / -23.95 / -21.68
Cholesterol lowering medication / -2 Log Likelihood / -148.16 / -130.98 / -212.40 / -207.16 / -204.96
AIC / -118.16 / -100.98 / -184.40 / -177.16 / -176.96
BIC / -84.43 / -67.25 / -152.72 / -143.43 / -145.28

Coefficients of generic proportion models: Acid blocking agents

generic proportion in NIHDI costs
Time / Effect / Estimate (s.e.)
intercept / -84.165 / (5.544)
Time / 2.925 / (0.191)
Time² / -0.014 / (0.0008)
Time³ / 0.00005 / (0.000003)
Jun99 / Intercept / 0.928 / (0.192)
Time / -1.842 / (0.192)
Nov99 / Intercept / -0.988 / (0.168)
Time / 0.040 / (0.044)
Dec99 / Intercept / -0.578 / (0.153)
Feb01 / Intercept / 1.717 / (0.091)
Time / 0.166 / (0.017)
Nov02 / Intercept / 0.514 / (0.076)
Time / 0.243 / (0.014)
Sep07 / Intercept / 0.099 / (0.071)
Time / -0.127 / (0.015)
generic proportion in co-payment costs
Time / Effect / Estimate (s.e.)
intercept / -81.829 / (5.447)
Time / 2.709 / (0.188)
Time² / -0.009 / (0.0008)
Time³ / 0.00002 / (0.000003)
Jun99 / Intercept / 0.996 / (0.188)
Time / -1.924 / (0.188)
Nov99 / Intercept / -1.316 / (0.125)
Time / -0.067 / (0.043)
Feb01 / Intercept / 1.439 / (0.209)
Time / 0.297 / (0.093)
May01 / Intercept / 0.457 / 0.137)
Time / -0.190 / (0.092)
Nov02 / Intercept / 0.518 / (0.082)
Time / 0.208 / (0.018)
Jun05 / Intercept / -0.334 / (0.070)
Time / 0.028 / (0.012)
generic proportion in packages sold
Time / Effect
intercept / -81.105 / (4.570)
Time / 2.710 / (0.158)
Time² / -0.010 / (0.0007)
Time³ / 0.00003 / (0.000003)
Jun99 / Intercept / 0.973 / (0.158)
Time / -1.875 / (0.158)
Nov99 / Intercept / -1.241 / (0.105)
Time / -0.041 / (0.036)
Feb01 / Intercept / 1.652 / (0.175)
Time / 0.266 / (0.078)
May01 / Intercept / 0.234 / (0.114)
Time / -0.176 / (0.077)
Nov02 / Intercept / 0.408 / (0.067)
Time / 0.188 / (0.011)
Sep07 / Intercept / 0.014 / (0.059)
Time / -0.077 / (0.012)
generic proportion in DDD sold
Time / Effect
intercept / -78.705 / (4.356)
Time / 2.553 / (0.151)
Time² / -0.007 / (0.0007)
Time³ / 0.000019 / (0.000002)
Jun99 / Intercept / 0.997 / (0.150)
Time / -1.896 / (0.150)
Nov99 / Intercept / -1.221 / (0.100)
Time / -0.058 / (0.035)
Feb01 / Intercept / 1.442 / (0.137)
Time / 0.305 / (0.047)
Jun01 / Intercept / 0.117 / (0.103)
Time / -0.273 / (0.047)
Nov02 / Intercept / 0.415 / (0.068)
Time / 0.177 / (0.014)
Jun05 / Intercept / -0.434 / (0.056)
Time / 0.019 / (0.010)
generic proportion in units sold
Time / Effect
intercept / -84.042 / (4.400)
Time / 2.732 / (0.152)
Time² / -0.007 / (0.0007)
Time³ / 0.00002 / (0.000002)
Jun99 / Intercept / 0.946 / (0.152)
Time / -2.080 / (0.152)
Nov99 / Intercept / -1.223 / (0.101)
Time / -0.063 / (0.035)
Feb01 / Intercept / 1.416 / (0.139)
Time / 0.308 / (0.048)
Jun01 / Intercept / 0.128 / (0.104)
Time / -0.283 / (0.048)
Nov02 / Intercept / 0.244 / (0.068)
Time / 0.167 / (0.015)
Jun05 / Intercept / -0.247 / (0.056)
Time / 0.019 / (0.010)

Model for cost per DDD: linear mixed model

Formula

(Co-payment cost / DDD)ij= β1,G + β1,B + β2,G*timei+ β2,B*timei+ β3,G*(NIHDI cost / DDD)ij

+ β3,B*(NIHDI cost / DDD)ij+ β4,G*timei*(NIHDI cost / DDD)ij

+ β4,B*timei*(NIHDI cost / DDD)ij+ β5,G*(NIHDI cost / DDD)ij² + β5,B*(NIHDI cost / DDD)ij²

+ βG(cp1)*I{timei>cp1} + βB(cp2)*I{timei>cp2} + βG(cp1)*I{timei>cp1}*(NIHDI cost / DDD)ij

+ βB(cp2)*I{timei>cp2}*(NIHDI cost / DDD)ij+…+ βG[cp(k-1)]*I{timei>cp(k-1)}+ βB(cpk)*I{timei>cpk}

+ βG[cp(k-1)]*I{timei>cp(k-1)}*(NIHDI cost / DDD)ij+ βB(cpk) *I{timei>cpk}*(NIHDI cost / DDD)ij

= Xjβ

Goodness of fit statistics

Statistic
Omeprazole / -2 Log Likelihood / -34257.1
AIC / -34243.1
BIC / -34224.8
Simvastatin / -2 Log Likelihood / -19820.5
AIC / -19806.5
BIC / -19793.6

Coefficients for the omeprazole model.

Table 2: Parameter estimates with corresponding standard errors for omeprazole. Estimates pointed out with an asterisk (*) are not significant (α = 0.05).

Generic / Brand
Time / Effect / Estimate (s.e.) / Time / Effect / Estimate (s.e.)
Jun02 / intercept / 0.298 / (0.070) / Jan97 / intercept / 0.860 / (0.239)
NIHDI cost / -0.599 / (0.165) / NIHDI cost / -0.635 / (0.261)
time / 0.00003* / (0.00007) / time / -0.001 / (0.0002)
(NIHDI cost)² / 0.199 / (0.081) / NIHDI cost * time / 0.001 / (0.0001)
Generic 1 / (NIHDI cost)² / 0.161 / (0.079)
Time / Estimate (s.e.) / Jun05 / intercept / 0.156 / (0.035)
Jun05 / intercept / -0.167 / (0.041) / NIHDI cost / 0.174 / (0.026)
NIHDI cost / 0.881 / (0.045) / Aug05 / intercept / -0.104 / (0.047)
Oct05 / intercept / 0.074 / (0.027) / NIHDI cost / -0.290 / (0.052)
NIHDI cost / -0.093 / (0.045) / Jun06 / NIHDI cost / -0.020 / (0.007)
Apr09 / NIHDI cost / -0.014 / (0.004) / Apr09 / intercept / -0.027 / (0.005)
Jun09 / intercept / -0.008 / (0.002)
Generic 2
Time / Effect / Estimate (s.e.)
Dec04 / intercept / -0.034 / (0.011)
NIHDI cost / 0.053 / (0.020)
Apr08 / intercept / -0.024 / (0.009)
NIHDI cost / 0.046 / (0.017)
Apr09 / intercept / -0.055 / (0.021)
NIHDI cost / 0.089 / (0.038)

(In the case of omeprazole, two subgroups were present among the generic products: some generic products exhibit a steep increase in co-payment cost per DDD around June 2005, while other generic products stay constant in price. These subgroups were analysed separately and the former of these subgroups is referred to as generic 1 in table 2, and the latter as generic 2. Generic products which entered the market after June 2005 are analysed with the largest subset, which is generic 1. It was not possible to determine what defines these subgroups.)

Coefficients for the simvastatin model.

Table 5: Parameter estimates with corresponding standard errors for simvastatin. Estimates pointed out with an asterisk (*) are not significant (α = 0.05).

Generic / Brand
Time / Effect / Estimate (s.e.) / Time / Effect / Estimate (s.e.)
Mar04 / intercept / 0.064 / (0.014) / Jan97 / intercept / 0.604 / (0.157)
NIHDI cost / 0.169 / (0.036) / NIHDI cost / -0.577 / (0.211)
time / -0.00007 / (0.00003) / time / -0.0003* / (0.0002)
(NIHDI cost)² / -0.054 / (0.025) / NIHDI cost * time / 0.0003 / (0.00005)
Generic 1 / (NIHDI cost)² / 0.181 / (0.067)
Time / Effect / Estimate (s.e.) / Jan00 / intercept / 0.144 / (0.020)
Jul04 / intercept / -0.117 / (0.014) / NIHDI cost / -0.081 / (0.010)
NIHDI cost / 0.288 / (0.024) / Jun03 / intercept / -0.378 / (0.083)
Oct05 / intercept / 0.040 / (0.008) / NIHDI cost / 0.307 / (0.062)
NIHDI cost / -0.054 / (0.019) / Dec05 / NIHDI cost / -0.058 / (0.003)
Generic 2 / Dec07 / NIHDI cost / -0.273 / (0.051)
Time / Effect / Estimate (s.e.) / brand 1
Dec03 / intercept / -0.023 / (0.012) / Time / Effect / Estimate (s.e.)
NIHDI cost / 0.042 / (0.016) / Dec02 / intercept / -0.121 / (0.028)
Oct05 / intercept / -0.004 / (0.001) / NIHDI cost / 0.115 / (0.017)
Dec07 / intercept / -0.029 / (0.007) / Brand 2
NIHDI cost / 0.075 / (0.022) / Time / Effect / Estimate (s.e.)
Jun05 / intercept / -0.125 / (0.038)
NIHDI cost / 0.199 / (0.039)

(Also in the group of simvastatin products, subgroups are detected. When all generic products are analysed as being one group, some products suffer a worse model-fit than the majority. These generic products are classified in subgroup generic 1, all other generic simvastatin products are in subgroup generic 2. Due to convergence issues, it was not possible to allow both generic groups to have separate intercepts, time and NIHDI cost parameters, only the change point effects are separate. The generic estimates in table 3 thus apply to all generic products, but the change points are listed separate for the two subgroups. Also among the five brand products there were remarkable changes in co-payment cost that were not present for all products. The products in subgroup brand 1 show an supplementary increase in December 2000, and subgroup 2 contains the products which have a decrease around June 2005, which all other brand products lack. This implies that the parameter estimates, including change point effects, in table 3 labelled as brand apply to all brand products, and that the subgroups brand 1 and brand 2 have additional change point parameters. Unfortunately, also for the simvastatin subgroups it was not possible to determine in which way they differ from each other. The partitioning in subgroups in listed in appendix B, table B.2.)