pima_dataset
2024-12-16
PIMA Indians Diabetes dataset
Riferences:
- Smith, J. W., Everhart, J. E., Dickson, W. C., Knowler, W. C. and
Johannes, R. S. (1988) Using the ADAP learning algorithm to forecast the
onset of diabetes mellitus. In Proceedings of the Symposium on Computer
Applications in Medical Care (Washington, 1988), ed. R. A. Greenes,
pp. 261–265. Los Alamitos, CA: IEEE Computer Society Press.
- Newman, D.J. & Hettich, S. & Blake, C.L. & Merz, C.J.
(1998). UCI Repository of machine learning databases http://www.ics.uci.edu/~mlearn/MLRepository.html.
Irvine, CA: University of California, Department of Information and
Computer Science.
The PIMA Indians dataset studies the predictors associated with the onset of diabetes. The sample includes 532 women from the Pima tribe of Native Americans.
npreg: number of pregnancies
glucose: glucose concentration in an oral glucose tolerance test.pressure: diastolic blood pressure (mm Hg).skin: triceps skin fold thickness (mm).bmi: body mass index (weight in kg)/(height in m)^2.pedigree: diabetes pedigree function (familiarity).age: age in years.diabetes:posornegfor diabetic according to WHO criteria.
In particular, we are interested in evaluating the effect of
npreg on the presence of diabetes
(diabetes = pos).
NOTE: Missing data has been removed for convenience of presentation. However, handling missing data (are they random? can I replace them with plausible values without distorting the results?) is a crucial aspect in the analysis of real data and an active area of research.
Observational Studies
In the case of observational studies like this (as opposed to randomized studies), it is essential to analyze the data taking into account possible confounders; it is necessary to include in our model all other variables that may influence the response (presence of diabetes).
It is easy to assume that variables such as glucose,
bmi, and pedigree are associated with (and
predictors of) diabetes.
Furthermore, variables strongly associated with the number of
pregnancies npreg, such as age (the
correlation between these two variables is 0.642), risk concealing or
magnifying the observed effect. For example, if age is positively
associated with the onset of diabetes, an association between the number
of pregnancies and diabetes will also be observed. However, this
association is spurious because women with a higher number of
pregnancies are, on average, also older.
rm(list=ls())
#### some libraries
############
# results reporting
library(gtsummary)
library(tidyverse)
## ── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
## ✔ dplyr 1.1.4 ✔ readr 2.1.5
## ✔ forcats 1.0.0 ✔ stringr 1.5.1
## ✔ ggplot2 3.5.1 ✔ tibble 3.2.1
## ✔ lubridate 1.9.3 ✔ tidyr 1.3.1
## ✔ purrr 1.0.2
## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag() masks stats::lag()
## ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors
library(GGally)
## Registered S3 method overwritten by 'GGally':
## method from
## +.gg ggplot2
library(knitr)
library(gridExtra)
##
## Caricamento pacchetto: 'gridExtra'
##
## Il seguente oggetto è mascherato da 'package:dplyr':
##
## combine
library(corrplot)
## corrplot 0.95 loaded
# models
library(MASS)
##
## Caricamento pacchetto: 'MASS'
##
## Il seguente oggetto è mascherato da 'package:dplyr':
##
## select
##
## Il seguente oggetto è mascherato da 'package:gtsummary':
##
## select
# palettes
options(ggplot2.continuous.colour="viridis")
options(ggplot2.continuous.fill = "viridis")
scale_colour_discrete <- scale_colour_viridis_d
scale_fill_discrete <- scale_fill_viridis_d
opts_chunk$set(echo=FALSE,
cache=FALSE,
prompt=FALSE,
# tidy=TRUE,
comment=NA,
message=FALSE,
warning=FALSE)
######################
#### core functions (to be dropped when the pima package is ready)
library(jointest)
##
## Caricamento pacchetto: 'jointest'
##
## Il seguente oggetto è mascherato da 'package:gridExtra':
##
## combine
##
## Il seguente oggetto è mascherato da 'package:dplyr':
##
## combine
##
## Il seguente oggetto è mascherato da 'package:stats':
##
## p.adjust
library(flipscores)
library(pima)
data("pimads")
pimads=na.omit(pimads)Exploratory Data Analysis
npreg glucose pressure skin
Min. : 0.000 Min. : 56.00 Min. : 24.00 Min. : 7.00
1st Qu.: 1.000 1st Qu.: 98.75 1st Qu.: 64.00 1st Qu.:22.00
Median : 2.000 Median :115.00 Median : 72.00 Median :29.00
Mean : 3.517 Mean :121.03 Mean : 71.51 Mean :29.18
3rd Qu.: 5.000 3rd Qu.:141.25 3rd Qu.: 80.00 3rd Qu.:36.00
Max. :17.000 Max. :199.00 Max. :110.00 Max. :99.00
bmi pedigree age diabetes
Min. :18.20 Min. :0.0850 Min. :21.00 neg:355
1st Qu.:27.88 1st Qu.:0.2587 1st Qu.:23.00 pos:177
Median :32.80 Median :0.4160 Median :28.00
Mean :32.89 Mean :0.5030 Mean :31.61
3rd Qu.:36.90 3rd Qu.:0.6585 3rd Qu.:38.00
Max. :67.10 Max. :2.4200 Max. :81.00 pimads %>% tbl_summary(by = diabetes) %>% add_p() %>%as_gt() %>% gt::as_raw_html()| Characteristic | neg N = 3551 |
pos N = 1771 |
p-value2 |
|---|---|---|---|
| 1 Median (Q1, Q3) | |||
| 2 Wilcoxon rank sum test | |||
#
#
p1<-ggplot(pimads, aes(x=diabetes, y=npreg, color=diabetes)) +
geom_boxplot()+theme_bw()
p2<-ggplot(pimads, aes(x=diabetes, y=glucose, color=diabetes)) +
geom_boxplot()+theme_bw()
grid.arrange(p1,p2,ncol=2)
pm <- ggpairs(pimads, columns = c(1:7),aes(color= diabetes),upper = list(continuous = wrap("cor", method = "spearman")))
pm
Very high correlation between skin and bmi (0.68) and between age and npreg (0.64).
Logistic Model
Observational study design:
- Outcome: Diabetes
- Determinant: number of pregnancies
- Confounders: glucose, pressure, skin, bmi, pedigree, age.
Let’s consider logistic regression
\[logit(\pi)=log(\pi/(1-\pi))=\beta_0+\beta_1\mbox{npreg}+\beta_2\mbox{glucose}+\beta_3\mbox{pressure}+\beta_4\mbox{skin}+\beta_5\mbox{bmi}+\beta_6\mbox{pedigree}+\beta_7\mbox{age}\] where \(\pi\) is the probability of diabetes, and therefore
(substituting \(\beta_0+\beta_1\mbox{npreg}+\beta_2\mbox{glucose}+\beta_3\mbox{pressure}+\beta_4\mbox{skin}+\beta_5\mbox{bmi}+\beta_6\mbox{pedigree}+\beta_7\mbox{age} = XB\)):
\[\mbox{diabetes} \sim Bernulli(\pi=expit(XB))\] \(expit()\) is the inverse of logit function: \[expit(XB)=\frac{e^{XB}}{1+e^{XB}}\]
MULTIVERSE ANALYSIS
In the analysis of real data, we do not know if (and it is not always plausible to assume that) the relationship of the predictors is linear. It might be appropriate to consider non-linear effects of the variables, such as the quadratic transformation (\(X^2\)) or the square root \(\sqrt{X}\). This could apply to all predictors.
Furthermore, in real data, we might look for observations that are too influential (leverage) or outliers. We could also evaluate different ways of imputing missing data or including interactions between variables, etc.
Let’s try some possible models.
Call:
glm(formula = diabetes ~ ., family = binomial, data = pimads)
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -9.554651 0.994217 -9.610 < 2e-16 ***
npreg 0.122517 0.043743 2.801 0.005097 **
glucose 0.035321 0.004244 8.322 < 2e-16 ***
pressure -0.007695 0.010314 -0.746 0.455602
skin 0.006774 0.014759 0.459 0.646242
bmi 0.082678 0.023334 3.543 0.000395 ***
pedigree 1.308708 0.364040 3.595 0.000324 ***
age 0.026375 0.014000 1.884 0.059581 .
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 676.79 on 531 degrees of freedom
Residual deviance: 466.32 on 524 degrees of freedom
AIC: 482.32
Number of Fisher Scoring iterations: 5
Call:
glm(formula = diabetes ~ . - age + sqrt(age), family = binomial,
data = pimads)
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -10.773675 1.198241 -8.991 < 2e-16 ***
npreg 0.108802 0.044573 2.441 0.014647 *
glucose 0.035115 0.004246 8.270 < 2e-16 ***
pressure -0.008995 0.010338 -0.870 0.384237
skin 0.006263 0.014816 0.423 0.672502
bmi 0.083496 0.023337 3.578 0.000347 ***
pedigree 1.294344 0.364344 3.553 0.000382 ***
sqrt(age) 0.397604 0.175404 2.267 0.023403 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 676.79 on 531 degrees of freedom
Residual deviance: 464.71 on 524 degrees of freedom
AIC: 480.71
Number of Fisher Scoring iterations: 5
Call:
glm(formula = diabetes ~ . - age + sqrt(age) - npreg + I(npreg^2),
family = binomial, data = pimads)
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -10.640101 1.209553 -8.797 < 2e-16 ***
glucose 0.035166 0.004257 8.261 < 2e-16 ***
pressure -0.009560 0.010338 -0.925 0.355053
skin 0.006217 0.014850 0.419 0.675485
bmi 0.080733 0.023284 3.467 0.000526 ***
pedigree 1.293180 0.365009 3.543 0.000396 ***
sqrt(age) 0.427970 0.166805 2.566 0.010297 *
I(npreg^2) 0.009252 0.003734 2.478 0.013213 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 676.79 on 531 degrees of freedom
Residual deviance: 464.25 on 524 degrees of freedom
AIC: 480.25
Number of Fisher Scoring iterations: 5All confounders are continuous. We decide to explore the following transformations for each predictor:
- Square root (\(sqrt(x)\))
- Identity (\(x\))
- Second-order polynomial (\(x+x^2\)) (or simply \(x^2\) for npreg)
########################################## possible specifications:
combinations_matrix=expand.grid(list(
c("npreg","sqrt(npreg)","I(npreg^2)"),
c("age","sqrt(age)","poly(age, 2)"),
c("glucose","sqrt(glucose)","poly(glucose, 2)"),
c("pedigree","sqrt(pedigree)","poly(pedigree, 2)"),
c("bmi","sqrt(bmi)","poly(bmi, 2)")))
combinations=apply(combinations_matrix,1,paste,collapse="+")Then we estimate all specified models (\(3^5=243\))
#################### one list of formulas for each model:
#######
formulas1=sapply(combinations,function(cmb) paste0("diabetes ~ ",cmb," + pressure + skin"))
mod1=glm(as.formula(formulas1[1]),data=pimads,family=binomial)
summary(mod1)
Call:
glm(formula = as.formula(formulas1[1]), family = binomial, data = pimads)
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -9.554651 0.994217 -9.610 < 2e-16 ***
npreg 0.122517 0.043743 2.801 0.005097 **
age 0.026375 0.014000 1.884 0.059581 .
glucose 0.035321 0.004244 8.322 < 2e-16 ***
pedigree 1.308708 0.364040 3.595 0.000324 ***
bmi 0.082678 0.023334 3.543 0.000395 ***
pressure -0.007695 0.010314 -0.746 0.455602
skin 0.006774 0.014759 0.459 0.646242
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 676.79 on 531 degrees of freedom
Residual deviance: 466.32 on 524 degrees of freedom
AIC: 482.32
Number of Fisher Scoring iterations: 5
#stimiamo tutti i modelli:
mods=lapply(formulas1, function(frml) update(mod1,formula. = as.formula(frml)))Using the function, we apply the PIMA procedure to the list of models within the list mods. And we correct for multiplicity (max-T).
library(pima)
#res=pima(mods,tested_coeffs = c("npreg","sqrt(npreg)","I(npreg^2)"),n_flips = 2000)
load(file="G:/Il mio Drive/lavorincorso/PIMA/talk_4M/pima_res_maxT.Rdata")
Correlations among coefficients in all model specifications
Some plots
Vulcano Plot
pima:::plot.pima(res,p.values="raw")
pima:::plot.pima(res,p.values="raw",x="Part. Cor",y="z value")
pima:::plot.pima(res,p.values="adjusted",x="Part. Cor",y="z value")
#plot.pima(res,p.values="raw",x="-log10(p)",y="-log10(p.adj.maxT)")
#summary(res) |> filter(p.adj.maxT<.05)Specification-Curve-like Plot
The Pima Tree
What characteristics do the significant models have?
Classification tree:
rpart(formula = p ~ ., data = comb_wide[, -c(1:2)], method = method)
Variables actually used in tree construction:
[1] age npreg
Root node error: 54/243 = 0.22222
n= 243
CP nsplit rel error xerror xstd
1 0.50 0 1 1 0.12001
2 0.01 2 0 0 0.00000
, , = FALSE
I(npreg^2) npreg sqrt(npreg)
age 0 27 27
poly(age, 2) 27 27 27
sqrt(age) 0 27 27
, , = TRUE
I(npreg^2) npreg sqrt(npreg)
age 27 0 0
poly(age, 2) 0 0 0
sqrt(age) 27 0 0The output consists of: - Tspace: table of test
statistics (observed in the first row) - summary_table:
summary of the results - mods: list of the 243 estimated
models (one for each specification) - info: information
about the specifications (auxiliary object)
Choose Your Model
Now we can consider the models whose p-values are still significant after correction.
We can take the one with the highest significance (minimum p-value) or the one that seems reasonable to us:
sel_model<-which(res$summary_table$p.adj<0.05)
sel_model
[1] 3 6 12 15 21 24 30 33 39 42 48 51 57 60 66 69 75 78 84
[20] 87 93 96 102 105 111 114 120 123 129 132 138 141 147 150 156 159 165 168
[39] 174 177 183 186 192 195 201 204 210 213 219 222 228 231 237 240
tab=res$summary_table
tab$.assign=NULL
tab[sel_model,]
Model
3 I(npreg^2)+age+glucose+pedigree+bmi
6 I(npreg^2)+sqrt(age)+glucose+pedigree+bmi
12 I(npreg^2)+age+sqrt(glucose)+pedigree+bmi
15 I(npreg^2)+sqrt(age)+sqrt(glucose)+pedigree+bmi
21 I(npreg^2)+age+poly(glucose, 2)+pedigree+bmi
24 I(npreg^2)+sqrt(age)+poly(glucose, 2)+pedigree+bmi
30 I(npreg^2)+age+glucose+sqrt(pedigree)+bmi
33 I(npreg^2)+sqrt(age)+glucose+sqrt(pedigree)+bmi
39 I(npreg^2)+age+sqrt(glucose)+sqrt(pedigree)+bmi
42 I(npreg^2)+sqrt(age)+sqrt(glucose)+sqrt(pedigree)+bmi
48 I(npreg^2)+age+poly(glucose, 2)+sqrt(pedigree)+bmi
51 I(npreg^2)+sqrt(age)+poly(glucose, 2)+sqrt(pedigree)+bmi
57 I(npreg^2)+age+glucose+poly(pedigree, 2)+bmi
60 I(npreg^2)+sqrt(age)+glucose+poly(pedigree, 2)+bmi
66 I(npreg^2)+age+sqrt(glucose)+poly(pedigree, 2)+bmi
69 I(npreg^2)+sqrt(age)+sqrt(glucose)+poly(pedigree, 2)+bmi
75 I(npreg^2)+age+poly(glucose, 2)+poly(pedigree, 2)+bmi
78 I(npreg^2)+sqrt(age)+poly(glucose, 2)+poly(pedigree, 2)+bmi
84 I(npreg^2)+age+glucose+pedigree+sqrt(bmi)
87 I(npreg^2)+sqrt(age)+glucose+pedigree+sqrt(bmi)
93 I(npreg^2)+age+sqrt(glucose)+pedigree+sqrt(bmi)
96 I(npreg^2)+sqrt(age)+sqrt(glucose)+pedigree+sqrt(bmi)
102 I(npreg^2)+age+poly(glucose, 2)+pedigree+sqrt(bmi)
105 I(npreg^2)+sqrt(age)+poly(glucose, 2)+pedigree+sqrt(bmi)
111 I(npreg^2)+age+glucose+sqrt(pedigree)+sqrt(bmi)
114 I(npreg^2)+sqrt(age)+glucose+sqrt(pedigree)+sqrt(bmi)
120 I(npreg^2)+age+sqrt(glucose)+sqrt(pedigree)+sqrt(bmi)
123 I(npreg^2)+sqrt(age)+sqrt(glucose)+sqrt(pedigree)+sqrt(bmi)
129 I(npreg^2)+age+poly(glucose, 2)+sqrt(pedigree)+sqrt(bmi)
132 I(npreg^2)+sqrt(age)+poly(glucose, 2)+sqrt(pedigree)+sqrt(bmi)
138 I(npreg^2)+age+glucose+poly(pedigree, 2)+sqrt(bmi)
141 I(npreg^2)+sqrt(age)+glucose+poly(pedigree, 2)+sqrt(bmi)
147 I(npreg^2)+age+sqrt(glucose)+poly(pedigree, 2)+sqrt(bmi)
150 I(npreg^2)+sqrt(age)+sqrt(glucose)+poly(pedigree, 2)+sqrt(bmi)
156 I(npreg^2)+age+poly(glucose, 2)+poly(pedigree, 2)+sqrt(bmi)
159 I(npreg^2)+sqrt(age)+poly(glucose, 2)+poly(pedigree, 2)+sqrt(bmi)
165 I(npreg^2)+age+glucose+pedigree+poly(bmi, 2)
168 I(npreg^2)+sqrt(age)+glucose+pedigree+poly(bmi, 2)
174 I(npreg^2)+age+sqrt(glucose)+pedigree+poly(bmi, 2)
177 I(npreg^2)+sqrt(age)+sqrt(glucose)+pedigree+poly(bmi, 2)
183 I(npreg^2)+age+poly(glucose, 2)+pedigree+poly(bmi, 2)
186 I(npreg^2)+sqrt(age)+poly(glucose, 2)+pedigree+poly(bmi, 2)
192 I(npreg^2)+age+glucose+sqrt(pedigree)+poly(bmi, 2)
195 I(npreg^2)+sqrt(age)+glucose+sqrt(pedigree)+poly(bmi, 2)
201 I(npreg^2)+age+sqrt(glucose)+sqrt(pedigree)+poly(bmi, 2)
204 I(npreg^2)+sqrt(age)+sqrt(glucose)+sqrt(pedigree)+poly(bmi, 2)
210 I(npreg^2)+age+poly(glucose, 2)+sqrt(pedigree)+poly(bmi, 2)
213 I(npreg^2)+sqrt(age)+poly(glucose, 2)+sqrt(pedigree)+poly(bmi, 2)
219 I(npreg^2)+age+glucose+poly(pedigree, 2)+poly(bmi, 2)
222 I(npreg^2)+sqrt(age)+glucose+poly(pedigree, 2)+poly(bmi, 2)
228 I(npreg^2)+age+sqrt(glucose)+poly(pedigree, 2)+poly(bmi, 2)
231 I(npreg^2)+sqrt(age)+sqrt(glucose)+poly(pedigree, 2)+poly(bmi, 2)
237 I(npreg^2)+age+poly(glucose, 2)+poly(pedigree, 2)+poly(bmi, 2)
240 I(npreg^2)+sqrt(age)+poly(glucose, 2)+poly(pedigree, 2)+poly(bmi, 2)
Coeff Estimate Score Std. Error z value Part. Cor p
3 I(npreg^2) 0.010284342 828.5898 294.4546 2.813981 0.1173409 0.0175
6 I(npreg^2) 0.009252495 729.6656 290.7851 2.509295 0.1046225 0.0305
12 I(npreg^2) 0.010248933 824.7275 294.4931 2.800498 0.1179712 0.0180
15 I(npreg^2) 0.009213771 725.6008 290.7617 2.495517 0.1051133 0.0315
21 I(npreg^2) 0.010284776 827.9660 294.2826 2.813506 0.1175084 0.0165
24 I(npreg^2) 0.009250739 728.7465 290.5778 2.507922 0.1048007 0.0315
30 I(npreg^2) 0.010365127 835.4487 293.8498 2.843115 0.1216682 0.0165
33 I(npreg^2) 0.009332483 736.3471 290.2739 2.536732 0.1085070 0.0290
39 I(npreg^2) 0.010328069 831.7298 293.9332 2.829656 0.1217861 0.0175
42 I(npreg^2) 0.009292547 732.4328 290.2992 2.523027 0.1085513 0.0300
48 I(npreg^2) 0.010365750 834.8552 293.6922 2.842620 0.1217585 0.0155
51 I(npreg^2) 0.009330844 735.4701 290.0854 2.535357 0.1085903 0.0295
57 I(npreg^2) 0.010321326 823.3681 291.5734 2.823880 0.1234264 0.0185
60 I(npreg^2) 0.009274219 724.2757 288.0526 2.514387 0.1098667 0.0305
66 I(npreg^2) 0.010290705 820.3633 291.7100 2.812256 0.1228721 0.0190
69 I(npreg^2) 0.009241731 721.1389 288.1354 2.502778 0.1093244 0.0315
75 I(npreg^2) 0.010324120 823.3493 291.4813 2.824707 0.1234626 0.0190
78 I(npreg^2) 0.009275896 724.0823 287.9471 2.514637 0.1098785 0.0310
84 I(npreg^2) 0.010309294 828.3333 294.1457 2.816065 0.1172593 0.0165
87 I(npreg^2) 0.009283474 729.9674 290.4487 2.513240 0.1046184 0.0300
93 I(npreg^2) 0.010275401 824.5302 294.1700 2.802904 0.1179327 0.0175
96 I(npreg^2) 0.009246288 725.9726 290.4117 2.499805 0.1051517 0.0315
102 I(npreg^2) 0.010307889 827.4362 293.9614 2.814778 0.1174791 0.0165
105 I(npreg^2) 0.009279555 728.7614 290.2263 2.511010 0.1048402 0.0310
111 I(npreg^2) 0.010388528 835.0963 293.5778 2.844548 0.1216394 0.0160
114 I(npreg^2) 0.009361567 736.5210 289.9714 2.539978 0.1085499 0.0290
120 I(npreg^2) 0.010353128 831.4352 293.6470 2.831411 0.1217942 0.0170
123 I(npreg^2) 0.009323273 732.6742 289.9829 2.526612 0.1086302 0.0300
129 I(npreg^2) 0.010387265 834.2444 293.4109 2.843263 0.1217473 0.0160
132 I(npreg^2) 0.009357752 735.3721 289.7710 2.537770 0.1086454 0.0290
138 I(npreg^2) 0.010346630 823.0843 291.3202 2.825360 0.1234557 0.0180
141 I(npreg^2) 0.009305583 724.5436 287.7658 2.517824 0.1099757 0.0305
147 I(npreg^2) 0.010317834 820.1426 291.4421 2.814084 0.1229349 0.0185
150 I(npreg^2) 0.009274886 721.4778 287.8339 2.506577 0.1094650 0.0310
156 I(npreg^2) 0.010348044 822.8654 291.2228 2.825553 0.1234661 0.0185
159 I(npreg^2) 0.009305681 724.1410 287.6523 2.517418 0.1099607 0.0310
165 I(npreg^2) 0.010279345 822.7593 293.6784 2.801565 0.1166379 0.0175
168 I(npreg^2) 0.009276460 726.5084 289.9579 2.505565 0.1042594 0.0295
174 I(npreg^2) 0.010248252 819.0198 293.6715 2.788897 0.1174101 0.0175
177 I(npreg^2) 0.009242613 722.6303 289.8895 2.492778 0.1048876 0.0310
183 I(npreg^2) 0.010274053 821.2777 293.4631 2.798572 0.1169836 0.0180
186 I(npreg^2) 0.009268397 724.7475 289.7025 2.501696 0.1045823 0.0315
192 I(npreg^2) 0.010355187 829.1480 293.1406 2.828500 0.1212268 0.0160
195 I(npreg^2) 0.009350588 732.6746 289.5062 2.530773 0.1083625 0.0285
201 I(npreg^2) 0.010322690 825.5339 293.1785 2.815806 0.1214546 0.0160
204 I(npreg^2) 0.009315640 728.9247 289.4858 2.517998 0.1085150 0.0300
210 I(npreg^2) 0.010350003 827.7414 292.9483 2.825555 0.1213809 0.0160
213 I(npreg^2) 0.009342635 730.9966 289.2781 2.526969 0.1084919 0.0295
219 I(npreg^2) 0.010316067 817.4943 290.9421 2.809818 0.1232769 0.0180
222 I(npreg^2) 0.009296274 720.9765 287.3627 2.508943 0.1099872 0.0305
228 I(npreg^2) 0.010289668 814.5561 291.0345 2.798830 0.1227980 0.0180
231 I(npreg^2) 0.009268392 717.9611 287.3999 2.498126 0.1095199 0.0315
237 I(npreg^2) 0.010315002 816.8861 290.8276 2.808833 0.1232429 0.0190
240 I(npreg^2) 0.009293729 720.2039 287.2299 2.507413 0.1099299 0.0300
p.adj.maxT
3 0.020
6 0.035
12 0.021
15 0.036
21 0.020
24 0.035
30 0.019
33 0.034
39 0.020
42 0.035
48 0.020
51 0.035
57 0.021
60 0.037
66 0.021
69 0.037
75 0.021
78 0.037
84 0.020
87 0.035
93 0.021
96 0.036
102 0.020
105 0.035
111 0.019
114 0.034
120 0.020
123 0.035
129 0.020
132 0.035
138 0.021
141 0.037
147 0.021
150 0.037
156 0.021
159 0.037
165 0.021
168 0.036
174 0.021
177 0.037
183 0.021
186 0.037
192 0.020
195 0.035
201 0.021
204 0.035
210 0.020
213 0.035
219 0.021
222 0.037
228 0.021
231 0.037
237 0.021
240 0.037Here is the translation of the provided markdown text:
Problematic Aspects
Of course, all these conclusions hold as long as the assumptions we make about the models we estimate are valid. It is appropriate to make adequacy assessments of the estimated models (e.g., diagnostic plots, fit indices).