A comprehensive statistical framework for designing and analyzing in vivo drug combination experiments.
You can install the development version of SynergyLMM from GitHub with:
# install.packages("pak")
pak::pak("RafRomB/SynergyLMM")
Or you can install the CRAN-released version with:
install.packages("SynergyLMM")
You can also use SynergyLMM directly in your browser at: https://synergylmm.uiocloud.no/
Example Use of SynergyLMMThis is a basic example which shows how to use SynergyLMM to analyze synergy in a 2-drug combination in vivo experiment.
We start by loading the data (in long format). We will use the example data provided in the package:
The first step is fitting the model from our data:
# Most simple model lmm <- lmmModel( data = grwth_data, sample_id = "subject", time = "Time", treatment = "Treatment", tumor_vol = "TumorVolume", trt_control = "Control", drug_a = "DrugA", drug_b = "DrugB", combination = "Combination" )
We can obtain the model estimates using:
lmmModel_estimates(lmm) #> Control sd_Control DrugA sd_DrugA DrugB sd_DrugB Combination #> 1 0.07855242 0.00322683 0.07491984 0.00322683 0.06306986 0.00322683 0.03487933 #> sd_Combination sd_ranef sd_resid #> 1 0.00322683 0.03946667 0.2124122
Bliss independence model
lmmSynergy(lmm, method = "Bliss")
#> $Contrasts
#> $Contrasts$Time3
#>
#> Hypothesis Estimate Std. Error z Pr(>|z|) S 2.5 % 97.5 %
#> b4=b2+b3-b1 -0.0649 0.0574 -1.13 0.258 2.0 -0.177 0.0476
#>
#>
#>
#> $Contrasts$Time6
#>
#> Hypothesis Estimate Std. Error z Pr(>|z|) S 2.5 % 97.5 %
#> b4=b2+b3-b1 -0.0326 0.0327 -0.996 0.319 1.6 -0.0968 0.0316
#>
#>
#>
#> $Contrasts$Time9
#>
#> Hypothesis Estimate Std. Error z Pr(>|z|) S 2.5 % 97.5 %
#> b4=b2+b3-b1 -0.053 0.0207 -2.56 0.0103 6.6 -0.0935 -0.0125
#>
#>
#>
#> $Contrasts$Time12
#>
#> Hypothesis Estimate Std. Error z Pr(>|z|) S 2.5 % 97.5 %
#> b4=b2+b3-b1 -0.039 0.0153 -2.55 0.0108 6.5 -0.069 -0.00901
#>
#>
#>
#> $Contrasts$Time15
#>
#> Hypothesis Estimate Std. Error z Pr(>|z|) S 2.5 % 97.5 %
#> b4=b2+b3-b1 -0.0401 0.0127 -3.15 0.00161 9.3 -0.065 -0.0152
#>
#>
#>
#> $Contrasts$Time18
#>
#> Hypothesis Estimate Std. Error z Pr(>|z|) S 2.5 % 97.5 %
#> b4=b2+b3-b1 -0.0366 0.0101 -3.62 <0.001 11.7 -0.0564 -0.0168
#>
#>
#>
#> $Contrasts$Time21
#>
#> Hypothesis Estimate Std. Error z Pr(>|z|) S 2.5 % 97.5 %
#> b4=b2+b3-b1 -0.0299 0.00846 -3.54 <0.001 11.3 -0.0465 -0.0134
#>
#>
#>
#> $Contrasts$Time24
#>
#> Hypothesis Estimate Std. Error z Pr(>|z|) S 2.5 % 97.5 %
#> b4=b2+b3-b1 -0.0278 0.00766 -3.63 <0.001 11.8 -0.0428 -0.0128
#>
#>
#>
#> $Contrasts$Time27
#>
#> Hypothesis Estimate Std. Error z Pr(>|z|) S 2.5 % 97.5 %
#> b4=b2+b3-b1 -0.0275 0.0069 -3.99 <0.001 13.9 -0.0411 -0.014
#>
#>
#>
#> $Contrasts$Time30
#>
#> Hypothesis Estimate Std. Error z Pr(>|z|) S 2.5 % 97.5 %
#> b4=b2+b3-b1 -0.0246 0.00645 -3.81 <0.001 12.8 -0.0372 -0.0119
#>
#>
#>
#>
#> $Synergy
#> Model Metric Estimate lwr upr pval Time
#> 1 Bliss CI 0.8229665 0.5872062 1.1533834 2.579032e-01 3
#> 2 Bliss CI 0.8221770 0.5593887 1.2084173 3.190110e-01 6
#> 3 Bliss CI 0.6208242 0.4312539 0.8937256 1.033643e-02 9
#> 4 Bliss CI 0.6263864 0.4371340 0.8975735 1.081171e-02 12
#> 5 Bliss CI 0.5478166 0.3769349 0.7961668 1.605259e-03 15
#> 6 Bliss CI 0.5176217 0.3623021 0.7395271 2.972850e-04 18
#> 7 Bliss CI 0.5332866 0.3764318 0.7555010 4.037916e-04 21
#> 8 Bliss CI 0.5132132 0.3580036 0.7357127 2.831685e-04 24
#> 9 Bliss CI 0.4754203 0.3300312 0.6848579 6.535614e-05 27
#> 10 Bliss CI 0.4786726 0.3275199 0.6995835 1.416473e-04 30
#> 11 Bliss SS 1.1313609 -0.8286031 3.0913249 2.579032e-01 3
#> 12 Bliss SS 0.9964923 -0.9634717 2.9564563 3.190110e-01 6
#> 13 Bliss SS 2.5643664 0.6044025 4.5243304 1.033643e-02 9
#> 14 Bliss SS 2.5487265 0.5887625 4.5086905 1.081171e-02 12
#> 15 Bliss SS 3.1549495 1.1949856 5.1149135 1.605259e-03 15
#> 16 Bliss SS 3.6176544 1.6576904 5.5776184 2.972850e-04 18
#> 17 Bliss SS 3.5375932 1.5776292 5.4975572 4.037916e-04 21
#> 18 Bliss SS 3.6302302 1.6702662 5.5901942 2.831685e-04 24
#> 19 Bliss SS 3.9925877 2.0326237 5.9525517 6.535614e-05 27
#> 20 Bliss SS 3.8052741 1.8453101 5.7652380 1.416473e-04 30
#>
#> $Estimates
#> Control sd_Control DrugA sd_DrugA DrugB sd_DrugB
#> 1 0.04981794 0.028702864 0.09505068 0.028702864 0.06953799 0.028702864
#> 2 0.07873777 0.016374072 0.07668022 0.016374072 0.07181158 0.016374072
#> 3 0.07679031 0.010327597 0.08295521 0.010327597 0.07551383 0.010327597
#> 4 0.08106700 0.007647412 0.08113043 0.007647412 0.06560025 0.007647412
#> 5 0.07886492 0.006358418 0.08014078 0.006358418 0.06433284 0.006358418
#> 6 0.07747072 0.005056304 0.08010240 0.005056304 0.06275579 0.005056304
#> 7 0.07725387 0.004231397 0.07627942 0.004231397 0.06183587 0.004231397
#> 8 0.07697505 0.003828178 0.07601450 0.003828178 0.06241830 0.003828178
#> 9 0.07721556 0.003448780 0.07629324 0.003448780 0.06200920 0.003448780
#> 10 0.07855242 0.003226830 0.07491984 0.003226830 0.06306986 0.003226830
#> Combination sd_Combination sd_ranef sd_resid Time
#> 1 0.04982413 0.028702864 0.88888751 0.08551689 3
#> 2 0.03712075 0.016374072 0.19732648 0.18726995 6
#> 3 0.02871125 0.010327597 0.11749473 0.19810677 9
#> 4 0.02668136 0.007647412 0.08599804 0.20531013 12
#> 5 0.02548772 0.006358418 0.07365316 0.20843087 15
#> 6 0.02880356 0.005056304 0.05767930 0.21206673 18
#> 7 0.03092350 0.004231397 0.04891680 0.21201738 21
#> 8 0.03366341 0.003828178 0.04623044 0.20908097 24
#> 9 0.03354777 0.003448780 0.04236597 0.20869435 27
#> 10 0.03487933 0.003226830 0.03946667 0.21241223 30
#>
#> $nsim
#> [1] 1000
#>
#> attr(,"SynergyLMM")
#> [1] "lmmSynergy"
Highest Single Agent model
lmmSynergy(lmm, method = "HSA")
#> $Contrasts
#> $Contrasts$Time3
#>
#> Hypothesis Estimate Std. Error z Pr(>|z|) S 2.5 % 97.5 %
#> b4=b3 -0.0197 0.0406 -0.486 0.627 0.7 -0.0993 0.0598
#>
#>
#>
#> $Contrasts$Time6
#>
#> Hypothesis Estimate Std. Error z Pr(>|z|) S 2.5 % 97.5 %
#> b4=b3 -0.0347 0.0232 -1.5 0.134 2.9 -0.0801 0.0107
#>
#>
#>
#> $Contrasts$Time9
#>
#> Hypothesis Estimate Std. Error z Pr(>|z|) S 2.5 % 97.5 %
#> b4=b3 -0.0468 0.0146 -3.2 0.00135 9.5 -0.0754 -0.0182
#>
#>
#>
#> $Contrasts$Time12
#>
#> Hypothesis Estimate Std. Error z Pr(>|z|) S 2.5 % 97.5 %
#> b4=b3 -0.0389 0.0108 -3.6 <0.001 11.6 -0.0601 -0.0177
#>
#>
#>
#> $Contrasts$Time15
#>
#> Hypothesis Estimate Std. Error z Pr(>|z|) S 2.5 % 97.5 %
#> b4=b3 -0.0388 0.00899 -4.32 <0.001 16.0 -0.0565 -0.0212
#>
#>
#>
#> $Contrasts$Time18
#>
#> Hypothesis Estimate Std. Error z Pr(>|z|) S 2.5 % 97.5 %
#> b4=b3 -0.034 0.00715 -4.75 <0.001 18.9 -0.048 -0.0199
#>
#>
#>
#> $Contrasts$Time21
#>
#> Hypothesis Estimate Std. Error z Pr(>|z|) S 2.5 % 97.5 %
#> b4=b3 -0.0309 0.00598 -5.17 <0.001 22.0 -0.0426 -0.0192
#>
#>
#>
#> $Contrasts$Time24
#>
#> Hypothesis Estimate Std. Error z Pr(>|z|) S 2.5 % 97.5 %
#> b4=b3 -0.0288 0.00541 -5.31 <0.001 23.1 -0.0394 -0.0181
#>
#>
#>
#> $Contrasts$Time27
#>
#> Hypothesis Estimate Std. Error z Pr(>|z|) S 2.5 % 97.5 %
#> b4=b3 -0.0285 0.00488 -5.84 <0.001 27.5 -0.038 -0.0189
#>
#>
#>
#> $Contrasts$Time30
#>
#> Hypothesis Estimate Std. Error z Pr(>|z|) S 2.5 % 97.5 %
#> b4=b3 -0.0282 0.00456 -6.18 <0.001 30.5 -0.0371 -0.0192
#>
#>
#>
#>
#> $Synergy
#> Model Metric Estimate lwr upr pval Time
#> 1 HSA CI 0.9425733 0.7424364 1.1966606 6.272090e-01 3
#> 2 HSA CI 0.8120893 0.6184990 1.0662733 1.341054e-01 6
#> 3 HSA CI 0.6562436 0.5071957 0.8490916 1.353136e-03 9
#> 4 HSA CI 0.6268633 0.4860749 0.8084303 3.199611e-04 12
#> 5 HSA CI 0.5584017 0.4286814 0.7273758 1.561099e-05 15
#> 6 HSA CI 0.5427317 0.4217207 0.6984663 2.053330e-06 18
#> 7 HSA CI 0.5224846 0.4084194 0.6684065 2.394746e-07 21
#> 8 HSA CI 0.5015173 0.3887649 0.6469709 1.088190e-07 24
#> 9 HSA CI 0.4637274 0.3582368 0.6002819 5.363760e-09 27
#> 10 HSA CI 0.4292500 0.3282301 0.5613609 6.512811e-10 30
#> 11 HSA SS 0.4856590 -1.4743050 2.4456229 6.272090e-01 3
#> 12 HSA SS 1.4981074 -0.4618566 3.4580714 1.341054e-01 6
#> 13 HSA SS 3.2044652 1.2445012 5.1644292 1.353136e-03 9
#> 14 HSA SS 3.5985787 1.6386147 5.5585426 3.199611e-04 12
#> 15 HSA SS 4.3198859 2.3599219 6.2798499 1.561099e-05 15
#> 16 HSA SS 4.7481035 2.7881395 6.7080675 2.053330e-06 18
#> 17 HSA SS 5.1657517 3.2057877 7.1257157 2.394746e-07 21
#> 18 HSA SS 5.3113453 3.3513813 7.2713092 1.088190e-07 24
#> 19 HSA SS 5.8354748 3.8755108 7.7954388 5.363760e-09 27
#> 20 HSA SS 6.1774914 4.2175274 8.1374554 6.512811e-10 30
#>
#> $Estimates
#> Control sd_Control DrugA sd_DrugA DrugB sd_DrugB
#> 1 0.04981794 0.028702864 0.09505068 0.028702864 0.06953799 0.028702864
#> 2 0.07873777 0.016374072 0.07668022 0.016374072 0.07181158 0.016374072
#> 3 0.07679031 0.010327597 0.08295521 0.010327597 0.07551383 0.010327597
#> 4 0.08106700 0.007647412 0.08113043 0.007647412 0.06560025 0.007647412
#> 5 0.07886492 0.006358418 0.08014078 0.006358418 0.06433284 0.006358418
#> 6 0.07747072 0.005056304 0.08010240 0.005056304 0.06275579 0.005056304
#> 7 0.07725387 0.004231397 0.07627942 0.004231397 0.06183587 0.004231397
#> 8 0.07697505 0.003828178 0.07601450 0.003828178 0.06241830 0.003828178
#> 9 0.07721556 0.003448780 0.07629324 0.003448780 0.06200920 0.003448780
#> 10 0.07855242 0.003226830 0.07491984 0.003226830 0.06306986 0.003226830
#> Combination sd_Combination sd_ranef sd_resid Time
#> 1 0.04982413 0.028702864 0.88888751 0.08551689 3
#> 2 0.03712075 0.016374072 0.19732648 0.18726995 6
#> 3 0.02871125 0.010327597 0.11749473 0.19810677 9
#> 4 0.02668136 0.007647412 0.08599804 0.20531013 12
#> 5 0.02548772 0.006358418 0.07365316 0.20843087 15
#> 6 0.02880356 0.005056304 0.05767930 0.21206673 18
#> 7 0.03092350 0.004231397 0.04891680 0.21201738 21
#> 8 0.03366341 0.003828178 0.04623044 0.20908097 24
#> 9 0.03354777 0.003448780 0.04236597 0.20869435 27
#> 10 0.03487933 0.003226830 0.03946667 0.21241223 30
#>
#> $nsim
#> [1] 1000
#>
#> attr(,"SynergyLMM")
#> [1] "lmmSynergy"
Response Additivity
set.seed(123) lmmSynergy(lmm, method = "RA", ra_nsim = 1000) #> Warning in lmmSynergy.explme(lmm, method = "RA", ra_nsim = 1000): p-values #> below p<1e-03 are approximated to 0. If you used method = 'RA' consider #> increasing 'nsim' value for more precise p-values.
#> $Contrasts
#> NULL
#>
#> $Synergy
#> Model Metric Estimate lwr upr pval Time
#> 1 RA CI 0.9089179 0.7813434 1.0846505 0.218 3
#> 2 RA CI 0.9055315 0.7542042 1.1327706 0.286 6
#> 3 RA CI 0.7754000 0.6498966 0.9550991 0.018 9
#> 4 RA CI 0.7779157 0.6425884 0.9785308 0.034 12
#> 5 RA CI 0.7176829 0.5841667 0.9180532 0.004 15
#> 6 RA CI 0.6859093 0.5612723 0.8732145 0.000 18
#> 7 RA CI 0.7011725 0.5672237 0.9063538 0.004 21
#> 8 RA CI 0.6796691 0.5420699 0.9005386 0.004 24
#> 9 RA CI 0.6443225 0.5094300 0.8750994 0.002 27
#> 10 RA CI 0.6594483 0.5041756 0.9381342 0.028 30
#> 11 RA SS 1.1273375 -0.8996513 3.0688249 0.218 3
#> 12 RA SS 0.9231803 -1.1026992 2.8852908 0.286 6
#> 13 RA SS 2.3885242 0.3929852 4.4088136 0.018 9
#> 14 RA SS 2.1559397 0.1725549 4.1256837 0.034 12
#> 15 RA SS 2.5810488 0.6353678 4.5520371 0.004 15
#> 16 RA SS 2.9266302 0.9515214 4.8819932 0.000 18
#> 17 RA SS 2.6486297 0.6548913 4.5838008 0.004 21
#> 18 RA SS 2.6435814 0.6478594 4.5814821 0.004 24
#> 19 RA SS 2.7286171 0.7507349 4.6869759 0.002 27
#> 20 RA SS 2.2569102 0.2982078 4.2338835 0.028 30
#>
#> $Estimates
#> Control sd_Control DrugA sd_DrugA DrugB
#> estimates_Time_3 0.04981794 0.028702864 0.09505068 0.028702864 0.06953799
#> estimates_Time_6 0.07873777 0.016374072 0.07668022 0.016374072 0.07181158
#> estimates_Time_9 0.07679031 0.010327597 0.08295521 0.010327597 0.07551383
#> estimates_Time_12 0.08106700 0.007647412 0.08113043 0.007647412 0.06560025
#> estimates_Time_15 0.07886492 0.006358418 0.08014078 0.006358418 0.06433284
#> estimates_Time_18 0.07747072 0.005056304 0.08010240 0.005056304 0.06275579
#> estimates_Time_21 0.07725387 0.004231397 0.07627942 0.004231397 0.06183587
#> estimates_Time_24 0.07697505 0.003828178 0.07601450 0.003828178 0.06241830
#> estimates_Time_27 0.07721556 0.003448780 0.07629324 0.003448780 0.06200920
#> estimates_Time_30 0.07855242 0.003226830 0.07491984 0.003226830 0.06306986
#> sd_DrugB Combination sd_Combination sd_ranef sd_resid
#> estimates_Time_3 0.028702864 0.04982413 0.028702864 0.88888751 0.08551689
#> estimates_Time_6 0.016374072 0.03712075 0.016374072 0.19732648 0.18726995
#> estimates_Time_9 0.010327597 0.02871125 0.010327597 0.11749473 0.19810677
#> estimates_Time_12 0.007647412 0.02668136 0.007647412 0.08599804 0.20531013
#> estimates_Time_15 0.006358418 0.02548772 0.006358418 0.07365316 0.20843087
#> estimates_Time_18 0.005056304 0.02880356 0.005056304 0.05767930 0.21206673
#> estimates_Time_21 0.004231397 0.03092350 0.004231397 0.04891680 0.21201738
#> estimates_Time_24 0.003828178 0.03366341 0.003828178 0.04623044 0.20908097
#> estimates_Time_27 0.003448780 0.03354777 0.003448780 0.04236597 0.20869435
#> estimates_Time_30 0.003226830 0.03487933 0.003226830 0.03946667 0.21241223
#> Time
#> estimates_Time_3 3
#> estimates_Time_6 6
#> estimates_Time_9 9
#> estimates_Time_12 12
#> estimates_Time_15 15
#> estimates_Time_18 18
#> estimates_Time_21 21
#> estimates_Time_24 24
#> estimates_Time_27 27
#> estimates_Time_30 30
#>
#> $nsim
#> [1] 1000
#>
#> attr(,"SynergyLMM")
#> [1] "lmmSynergy"
Using Robust Estimates
lmmSynergy(lmm, method = "Bliss", robust = TRUE) #> Registered S3 method overwritten by 'clubSandwich': #> method from #> bread.mlm sandwich
#> $Contrasts
#> $Contrasts$Time3
#>
#> Hypothesis Estimate Std. Error z Pr(>|z|) S 2.5 % 97.5 %
#> b4=b2+b3-b1 -0.0649 0.0574 -1.13 0.258 2.0 -0.177 0.0476
#>
#>
#>
#> $Contrasts$Time6
#>
#> Hypothesis Estimate Std. Error z Pr(>|z|) S 2.5 % 97.5 %
#> b4=b2+b3-b1 -0.0326 0.0327 -0.996 0.319 1.6 -0.0968 0.0316
#>
#>
#>
#> $Contrasts$Time9
#>
#> Hypothesis Estimate Std. Error z Pr(>|z|) S 2.5 % 97.5 %
#> b4=b2+b3-b1 -0.053 0.0207 -2.56 0.0103 6.6 -0.0935 -0.0125
#>
#>
#>
#> $Contrasts$Time12
#>
#> Hypothesis Estimate Std. Error z Pr(>|z|) S 2.5 % 97.5 %
#> b4=b2+b3-b1 -0.039 0.0153 -2.55 0.0108 6.5 -0.069 -0.00901
#>
#>
#>
#> $Contrasts$Time15
#>
#> Hypothesis Estimate Std. Error z Pr(>|z|) S 2.5 % 97.5 %
#> b4=b2+b3-b1 -0.0401 0.0127 -3.15 0.00161 9.3 -0.065 -0.0152
#>
#>
#>
#> $Contrasts$Time18
#>
#> Hypothesis Estimate Std. Error z Pr(>|z|) S 2.5 % 97.5 %
#> b4=b2+b3-b1 -0.0366 0.0101 -3.62 <0.001 11.7 -0.0564 -0.0168
#>
#>
#>
#> $Contrasts$Time21
#>
#> Hypothesis Estimate Std. Error z Pr(>|z|) S 2.5 % 97.5 %
#> b4=b2+b3-b1 -0.0299 0.00846 -3.54 <0.001 11.3 -0.0465 -0.0134
#>
#>
#>
#> $Contrasts$Time24
#>
#> Hypothesis Estimate Std. Error z Pr(>|z|) S 2.5 % 97.5 %
#> b4=b2+b3-b1 -0.0278 0.00766 -3.63 <0.001 11.8 -0.0428 -0.0128
#>
#>
#>
#> $Contrasts$Time27
#>
#> Hypothesis Estimate Std. Error z Pr(>|z|) S 2.5 % 97.5 %
#> b4=b2+b3-b1 -0.0275 0.0069 -3.99 <0.001 13.9 -0.0411 -0.014
#>
#>
#>
#> $Contrasts$Time30
#>
#> Hypothesis Estimate Std. Error z Pr(>|z|) S 2.5 % 97.5 %
#> b4=b2+b3-b1 -0.0246 0.00645 -3.81 <0.001 12.8 -0.0372 -0.0119
#>
#>
#>
#>
#> $Synergy
#> Model Metric Estimate lwr upr pval Time
#> 1 Bliss CI 0.8229665 0.5872062 1.1533834 2.579032e-01 3
#> 2 Bliss CI 0.8221770 0.5593887 1.2084173 3.190110e-01 6
#> 3 Bliss CI 0.6208242 0.4312539 0.8937256 1.033643e-02 9
#> 4 Bliss CI 0.6263864 0.4371340 0.8975735 1.081171e-02 12
#> 5 Bliss CI 0.5478166 0.3769351 0.7961665 1.605239e-03 15
#> 6 Bliss CI 0.5176217 0.3623022 0.7395270 2.972826e-04 18
#> 7 Bliss CI 0.5332866 0.3764318 0.7555010 4.037908e-04 21
#> 8 Bliss CI 0.5132132 0.3580036 0.7357127 2.831685e-04 24
#> 9 Bliss CI 0.4754203 0.3300312 0.6848579 6.535614e-05 27
#> 10 Bliss CI 0.4786726 0.3275199 0.6995835 1.416473e-04 30
#> 11 Bliss SS 1.1313609 -0.8286031 3.0913249 2.579032e-01 3
#> 12 Bliss SS 0.9964923 -0.9634717 2.9564563 3.190110e-01 6
#> 13 Bliss SS 2.5643665 0.6044025 4.5243305 1.033643e-02 9
#> 14 Bliss SS 2.5487265 0.5887625 4.5086905 1.081171e-02 12
#> 15 Bliss SS 3.1549531 1.1949891 5.1149171 1.605239e-03 15
#> 16 Bliss SS 3.6176565 1.6576925 5.5776205 2.972826e-04 18
#> 17 Bliss SS 3.5375938 1.5776298 5.4975577 4.037908e-04 21
#> 18 Bliss SS 3.6302303 1.6702663 5.5901942 2.831685e-04 24
#> 19 Bliss SS 3.9925877 2.0326237 5.9525517 6.535614e-05 27
#> 20 Bliss SS 3.8052741 1.8453101 5.7652380 1.416473e-04 30
#>
#> $Estimates
#> Control sd_Control DrugA sd_DrugA DrugB sd_DrugB
#> 1 0.04981794 0.036617165 0.09505068 0.015324541 0.06953799 0.029394978
#> 2 0.07873777 0.016727786 0.07668022 0.015168248 0.07181158 0.014983429
#> 3 0.07679031 0.011169651 0.08295521 0.007304339 0.07551383 0.008607233
#> 4 0.08106700 0.007735252 0.08113043 0.005786250 0.06560025 0.006345697
#> 5 0.07886492 0.007185241 0.08014078 0.004999149 0.06433284 0.005545836
#> 6 0.07747072 0.006050362 0.08010240 0.002986071 0.06275579 0.004327717
#> 7 0.07725387 0.005157847 0.07627942 0.003195873 0.06183587 0.003236268
#> 8 0.07697505 0.004809212 0.07601450 0.002950251 0.06241830 0.002766731
#> 9 0.07721556 0.004502935 0.07629324 0.002757550 0.06200920 0.002354660
#> 10 0.07855242 0.004038033 0.07491984 0.002581041 0.06306986 0.002520362
#> Combination sd_Combination sd_ranef sd_resid Time
#> 1 0.04982413 0.029252256 0.88888751 0.08551689 3
#> 2 0.03712075 0.018385952 0.19732648 0.18726995 6
#> 3 0.02871125 0.013207501 0.11749473 0.19810677 9
#> 4 0.02668136 0.010017432 0.08599804 0.20531013 12
#> 5 0.02548772 0.007371709 0.07365316 0.20843087 15
#> 6 0.02880356 0.006165393 0.05767930 0.21206673 18
#> 7 0.03092350 0.004932385 0.04891680 0.21201738 21
#> 8 0.03366341 0.004374068 0.04623044 0.20908097 24
#> 9 0.03354777 0.003761834 0.04236597 0.20869435 27
#> 10 0.03487933 0.003511413 0.03946667 0.21241223 30
#>
#> $nsim
#> [1] 1000
#>
#> attr(,"SynergyLMM")
#> [1] "lmmSynergy"
We can perform the model diagnostics using the following functions:
Random Effects
#>
#> Normality Test of Random Effects
#> $Time
#>
#> Title:
#> Shapiro - Wilk Normality Test
#>
#> Test Results:
#> STATISTIC:
#> W: 0.9672
#> P VALUE:
#> 0.4269
#>
#> Description:
#> Normality Test of Time random effects
#>
#>
#> Normalized Residuals Levene Homoscedasticity Test by Sample
#> Levene's Test for Homogeneity of Variance (center = median)
#> Df F value Pr(>F)
#> group 31 0.874 0.6633
#> 288
#>
#> Normalized Residuals Fligner-Killeen Homoscedasticity Test by Sample
#>
#> Fligner-Killeen test of homogeneity of variances
#>
#> data: normalized_resid by SampleID
#> Fligner-Killeen:med chi-squared = 28.521, df = 31, p-value = 0.5942
Residuals Diagnostics
#>
#> Normalized Residuals Normality Test
#>
#> Title:
#> Shapiro - Wilk Normality Test
#>
#> Test Results:
#> STATISTIC:
#> W: 0.9895
#> P VALUE:
#> 0.02154
#>
#>
#> Normalized Residuals Levene Homoscedasticity Test by Time
#> Levene's Test for Homogeneity of Variance (center = median)
#> Df F value Pr(>F)
#> group 9 0.4714 0.8934
#> 310
#>
#> Normalized Residuals Fligner-Killeen Homoscedasticity Test by Time
#>
#> Fligner-Killeen test of homogeneity of variances
#>
#> data: normalized_resid by as.factor(Time)
#> Fligner-Killeen:med chi-squared = 3.8143, df = 9, p-value = 0.9232
#>
#>
#> Normalized Residuals Levene Homoscedasticity Test by Treatment
#> Levene's Test for Homogeneity of Variance (center = median)
#> Df F value Pr(>F)
#> group 3 0.5772 0.6304
#> 316
#>
#> Normalized Residuals Fligner-Killeen Homoscedasticity Test by Treatment
#>
#> Fligner-Killeen test of homogeneity of variances
#>
#> data: normalized_resid by Treatment
#> Fligner-Killeen:med chi-squared = 1.5405, df = 3, p-value = 0.673
#>
#>
#> Outlier observations
#> SampleID Time Treatment TV RTV logRTV TV0
#> 16 2 18 Control 512.1195 2.3886998 0.8707492 214.3926
#> 41 5 3 Control 197.1398 0.7938990 -0.2307990 248.3185
#> 51 6 3 Control 175.4958 0.6732778 -0.3955972 260.6588
#> 65 7 15 Control 357.3550 1.6397816 0.4945630 217.9284
#> 113 12 9 DrugA 514.0043 3.1708889 1.1540120 162.1010
#> 122 13 6 DrugA 179.3938 0.8896325 -0.1169469 201.6493
#> 135 14 15 DrugA 811.5854 5.8214346 1.7615467 139.4133
#> 149 15 27 DrugA 2182.0193 11.1771236 2.4138692 195.2219
#> 182 19 6 DrugB 425.3002 2.4226262 0.8848522 175.5534
#> 221 23 3 DrugB 187.7751 0.7226499 -0.3248305 259.8424
#> 243 25 9 Combination 185.9586 0.6374488 -0.4502813 291.7232
#> 272 28 6 Combination 317.8079 2.1773052 0.7780880 145.9639
#> 284 29 12 Combination 211.5939 0.8481839 -0.1646578 249.4670
#> 293 30 9 Combination 209.9097 0.8877339 -0.1190832 236.4557
#> 301 31 3 Combination 171.9011 0.7141648 -0.3366415 240.7023
#> 305 31 15 Combination 214.8480 0.8925879 -0.1136302 240.7023
#> 314 32 12 Combination 209.8804 0.8805177 -0.1272452 238.3602
#> normalized_resid
#> 16 -2.175456
#> 41 -2.107815
#> 51 -2.796859
#> 65 -2.948355
#> 113 2.013734
#> 122 -2.439570
#> 135 2.306236
#> 149 2.147614
#> 182 2.178362
#> 221 -2.256830
#> 243 -3.153712
#> 272 2.352650
#> 284 -2.467379
#> 293 -2.037821
#> 301 -2.012043
#> 305 -2.670916
#> 314 -2.113725
Observed versus Predicted Values
#> # Indices of model performance
#>
#> AIC | AICc | BIC | R2 (cond.) | R2 (marg.) | RMSE | Sigma
#> ------------------------------------------------------------------
#> 22.778 | 23.046 | 45.388 | 0.898 | 0.898 | 0.204 | 0.212
Influential Diagnostics
log likelihood displacements
logLikSubjectDisplacements(lmm) #> [1] "Outliers with Log Likelihood displacement greater than: 0.534" #> 6 8 25 28 #> 0.6308934 1.1633552 0.5465657 0.5354518
Cook’s distances
CookDistance(lmm) #> [1] "Subjects with Cook's distance greater than: 0.578" #> 8 #> 0.8326171
Post-Hoc Power Analysis
set.seed(123) PostHocPwr(lmm, method = "Bliss", time = 30) #> [1] 0.959
A Priori Power Analysis
We will estimate the effect of sample size on statistical power based on the estimates from the model:
# Vector with different sample sizes per group npg <- 3:15 # Obtain model estimates (lmmestim <- lmmModel_estimates(lmm)) #> Control sd_Control DrugA sd_DrugA DrugB sd_DrugB Combination #> 1 0.07855242 0.00322683 0.07491984 0.00322683 0.06306986 0.00322683 0.03487933 #> sd_Combination sd_ranef sd_resid #> 1 0.00322683 0.03946667 0.2124122 # Obtain time points (timepoints <- unique(lmm$dt1$Time)) #> [1] 0 3 6 9 12 15 18 21 24 27 30 # Calculate power depending on sample size per group PwrSampleSize( npg = npg, time = timepoints, grwrControl = round(lmmestim$Control,3), grwrA = round(lmmestim$DrugA,3), grwrB = round(lmmestim$DrugB,3), grwrComb = round(lmmestim$Combination,3), sd_ranef = round(lmmestim$sd_ranef,3), sgma = round(lmmestim$sd_resid,3), method = "Bliss" )
#> N Power
#> 1 3 0.6277188
#> 2 4 0.7533191
#> 3 5 0.8415526
#> 4 6 0.9008031
#> 5 7 0.9392242
#> 6 8 0.9634473
#> 7 9 0.9783671
#> 8 10 0.9873773
#> 9 11 0.9927269
#> 10 12 0.9958564
#> 11 13 0.9976633
#> 12 14 0.9986944
#> 13 15 0.9992767
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