TBIPFM_Qs_FA_K23Grant
Ash Rajesh
1/4/2024
References
- Burnham & Anderson, Model Selection and Multi-Model Inference: A Practical Information-Theoretic Approach, and Burnham & Anderson, “Multimodel Inference: Understanding AIC and BIC in Model Selection” (2004, Sociological Methods & Research)
- https://stats.stackexchange.com/questions/409537/how-to-include-an-interaction-with-a-quadratic-term
- https://stackoverflow.com/questions/42764028/fitting-a-quadratic-curve-in-ggplot
- https://stackoverflow.com/questions/24342162/regression-tables-in-markdown-format-for-flexible-use-in-r-markdown-v2
- https://stats.stackexchange.com/questions/282686/what-percentage-decrease-in-aic-is-significant
- https://stats.stackexchange.com/questions/84076/negative-values-for-aic-in-general-mixed-model
- https://uedufy.com/how-to-run-multiple-moderation-analysis-in-r-like-a-pro/
- https://rdrr.io/cran/interactions/man/interact_plot.html
- https://interactions.jacob-long.com/reference/interact_plot.html
Participants
TBIPFM01
Variables Over Time
Depression
MODEL
model1 <- lm(Dep_BDI ~ Time.since.injury, data=tbi1)
model2 <- lm(Dep_BDI ~ Time.since.injury + I(Time.since.injury^2), data=tbi1)
SUMMARY
| Dep_LinearModel | Dep_QuadraticModel | |||||
|---|---|---|---|---|---|---|
| Predictors | Estimates | CI | p | Estimates | CI | p |
| (Intercept) | 25.99 | 24.57 – 27.41 | <0.001 | 35.84 | 34.19 – 37.48 | <0.001 |
| Time since injury | -0.02 | -0.03 – -0.01 | 0.004 | -0.35 | -0.40 – -0.31 | <0.001 |
| Time since injury^2 | 0.00 | 0.00 – 0.00 | <0.001 | |||
| Observations | 486 | 486 | ||||
| R2 / R2 adjusted | 0.017 / 0.015 | 0.368 / 0.365 | ||||
ANOVA COMPARISON
| Res.Df | RSS | Df | Sum of Sq | F | Pr(>F) |
|---|---|---|---|---|---|
| 484 | 32623 | ||||
| 483 | 20992 | 1 | 11631 | 267.6 | 3.497e-48 |
AIC
AIC for model 1 (i.e., linear model) 3430
AIC for model 2 (i.e.,quadratic model) 3217
INTERPRETATION
- An ANOVA comparison suggests that there is a significant difference in the linear versus quadratic fit (p< 10^-29)
- The R-squared value of the linear model is 0.12. The adjusted R-squared value of the quadratic model is 0.32, suggesting more variance is explained by the second model.
- To ensure that that quadratic model is not better due to over-fitting, we also determined the AIC for each model. For the linear model, the AIC is 3152. For the quadratic model, the AIC is 3028. The ΔAIC is > 10, providing strong evidence that the model with lower AIC (i.e., the quadratic model) is a better fit.
Anxiety
MODEL
model1 <- lm(Anx_BAI ~ Time.since.injury, data=tbi1)
model2 <- lm(Anx_BAI ~ Time.since.injury + I(Time.since.injury^2), data=tbi1)
SUMMARY
| Anx_LinearModel | Anx_QuadraticModel | |||||
|---|---|---|---|---|---|---|
| Predictors | Estimates | CI | p | Estimates | CI | p |
| (Intercept) | 17.06 | 16.11 – 18.00 | <0.001 | 24.11 | 23.07 – 25.14 | <0.001 |
| Time since injury | -0.07 | -0.08 – -0.06 | <0.001 | -0.31 | -0.33 – -0.28 | <0.001 |
| Time since injury^2 | 0.00 | 0.00 – 0.00 | <0.001 | |||
| Observations | 486 | 486 | ||||
| R2 / R2 adjusted | 0.312 / 0.311 | 0.598 / 0.597 | ||||
ANOVA COMPARISON
| Res.Df | RSS | Df | Sum of Sq | F | Pr(>F) |
|---|---|---|---|---|---|
| 484 | 14325 | ||||
| 483 | 8364 | 1 | 5961 | 344.2 | 2.063e-58 |
AIC
AIC for model 1 (i.e., linear model) 3030
AIC for model 2 (i.e.,quadratic model) 2770
INTERPRETATION
- An ANOVA comparison suggests that there is a significant difference in the linear versus quadratic fit (p< 10^-52)
- The R-squared value of the linear model is 0.35. The adjusted R-squared value of the quadratic model is 0.60, suggesting more variance is explained by the second model.
- To ensure that that quadratic model is not better due to over-fitting, we also determined the AIC for each model. For the linear model, the AIC is 2911. For the quadratic model, the AIC is 2678. The ΔAIC is >10, providing strong evidence that the model with lower AIC (i.e., the quadratic model) is a better fit.
FA (without tract info)
MODEL
model1 <- lm(FA ~ Time.since.injury, data=tbi1)
model2 <- lm(FA ~ Time.since.injury + I(Time.since.injury^2), data=tbi1)
SUMMARY
| FA_LinearModel | FA_QuadraticModel | |||||
|---|---|---|---|---|---|---|
| Predictors | Estimates | CI | p | Estimates | CI | p |
| (Intercept) | 0.54 | 0.53 – 0.55 | <0.001 | 0.54 | 0.53 – 0.56 | <0.001 |
| Time since injury | -0.00 | -0.00 – 0.00 | 0.396 | -0.00 | -0.00 – 0.00 | 0.397 |
| Time since injury^2 | 0.00 | -0.00 – 0.00 | 0.516 | |||
| Observations | 486 | 486 | ||||
| R2 / R2 adjusted | 0.001 / -0.001 | 0.002 / -0.002 | ||||
ANOVA COMPARISON
| Res.Df | RSS | Df | Sum of Sq | F | Pr(>F) |
|---|---|---|---|---|---|
| 484 | 2.254 | ||||
| 483 | 2.252 | 1 | 0.001968 | 0.4222 | 0.5161 |
AIC
AIC for model 1 (i.e., linear model) -1226
AIC for model 2 (i.e.,quadratic model) -1225
INTERPRETATION
- FA over Time since injury (without tract info) is not significantly different over time.
FA by Tract and Time
MODEL BY TRACT AND TIME
model1 <- lm(FA ~ Tract.ID + Time.since.injury + Tract.ID*Time.since.injury, data=tbi1)
model2 <- lm(FA ~ Tract.ID + Time.since.injury + I(Time.since.injury^2) + Tract.ID:Time.since.injury + Tract.ID:I(Time.since.injury^2), data=tbi1)
SUMMARY FOR MODEL BY TRACT AND TIME
| FA_LinearModel | FA_QuadraticModel | |||||
|---|---|---|---|---|---|---|
| Predictors | Estimates | CI | p | Estimates | CI | p |
| (Intercept) | 0.68 | 0.68 – 0.68 | <0.001 | 0.69 | 0.68 – 0.69 | <0.001 |
| Tract ID [forceps minor] | -0.09 | -0.10 – -0.08 | <0.001 | -0.09 | -0.10 – -0.08 | <0.001 |
| Tract ID [left atr] | -0.18 | -0.19 – -0.18 | <0.001 | -0.19 | -0.19 – -0.18 | <0.001 |
| Tract ID [left cab] | -0.26 | -0.26 – -0.25 | <0.001 | -0.26 | -0.27 – -0.26 | <0.001 |
| Tract ID [left cing] | -0.03 | -0.03 – -0.02 | <0.001 | -0.03 | -0.04 – -0.02 | <0.001 |
| Tract ID [left cst] | -0.12 | -0.13 – -0.11 | <0.001 | -0.12 | -0.13 – -0.11 | <0.001 |
| Tract ID [left ilf] | -0.14 | -0.14 – -0.13 | <0.001 | -0.14 | -0.15 – -0.13 | <0.001 |
| Tract ID [left slfp] | -0.16 | -0.17 – -0.16 | <0.001 | -0.16 | -0.17 – -0.16 | <0.001 |
| Tract ID [left slft] | -0.13 | -0.13 – -0.12 | <0.001 | -0.13 | -0.14 – -0.12 | <0.001 |
| Tract ID [left unc] | -0.18 | -0.18 – -0.17 | <0.001 | -0.18 | -0.19 – -0.17 | <0.001 |
| Tract ID [right atr] | -0.19 | -0.19 – -0.18 | <0.001 | -0.19 | -0.20 – -0.18 | <0.001 |
| Tract ID [right cab] | -0.25 | -0.26 – -0.24 | <0.001 | -0.26 | -0.27 – -0.25 | <0.001 |
| Tract ID [right cing] | -0.01 | -0.02 – -0.01 | <0.001 | -0.01 | -0.02 – -0.01 | 0.001 |
| Tract ID [right cst] | -0.12 | -0.12 – -0.11 | <0.001 | -0.12 | -0.13 – -0.11 | <0.001 |
| Tract ID [right ilf] | -0.14 | -0.14 – -0.13 | <0.001 | -0.14 | -0.15 – -0.13 | <0.001 |
| Tract ID [right slfp] | -0.18 | -0.19 – -0.17 | <0.001 | -0.18 | -0.19 – -0.17 | <0.001 |
| Tract ID [right slft] | -0.16 | -0.16 – -0.15 | <0.001 | -0.16 | -0.17 – -0.15 | <0.001 |
| Tract ID [right unc] | -0.18 | -0.19 – -0.17 | <0.001 | -0.19 | -0.19 – -0.18 | <0.001 |
| Time since injury | -0.00 | -0.00 – -0.00 | 0.001 | -0.00 | -0.00 – -0.00 | <0.001 |
|
Tract ID [forceps minor] × Time since injury |
-0.00 | -0.00 – 0.00 | 0.950 | 0.00 | -0.00 – 0.00 | 0.500 |
|
Tract ID [left atr] × Time since injury |
0.00 | -0.00 – 0.00 | 0.081 | 0.00 | -0.00 – 0.00 | 0.083 |
|
Tract ID [left cab] × Time since injury |
0.00 | 0.00 – 0.00 | 0.011 | 0.00 | 0.00 – 0.00 | 0.001 |
|
Tract ID [left cing] × Time since injury |
0.00 | -0.00 – 0.00 | 0.518 | 0.00 | -0.00 – 0.00 | 0.401 |
|
Tract ID [left cst] × Time since injury |
0.00 | -0.00 – 0.00 | 0.200 | 0.00 | -0.00 – 0.00 | 0.236 |
|
Tract ID [left ilf] × Time since injury |
0.00 | -0.00 – 0.00 | 0.051 | 0.00 | -0.00 – 0.00 | 0.356 |
|
Tract ID [left slfp] × Time since injury |
0.00 | -0.00 – 0.00 | 0.054 | 0.00 | -0.00 – 0.00 | 0.135 |
|
Tract ID [left slft] × Time since injury |
0.00 | -0.00 – 0.00 | 0.122 | 0.00 | -0.00 – 0.00 | 0.136 |
|
Tract ID [left unc] × Time since injury |
0.00 | 0.00 – 0.00 | 0.032 | 0.00 | -0.00 – 0.00 | 0.148 |
|
Tract ID [right atr] × Time since injury |
-0.00 | -0.00 – 0.00 | 0.472 | 0.00 | -0.00 – 0.00 | 0.871 |
|
Tract ID [right cab] × Time since injury |
0.00 | -0.00 – 0.00 | 0.096 | 0.00 | 0.00 – 0.00 | <0.001 |
|
Tract ID [right cing] × Time since injury |
-0.00 | -0.00 – -0.00 | 0.001 | -0.00 | -0.00 – 0.00 | 0.469 |
|
Tract ID [right cst] × Time since injury |
0.00 | -0.00 – 0.00 | 0.150 | 0.00 | -0.00 – 0.00 | 0.121 |
|
Tract ID [right ilf] × Time since injury |
-0.00 | -0.00 – 0.00 | 0.708 | 0.00 | -0.00 – 0.00 | 0.269 |
|
Tract ID [right slfp] × Time since injury |
0.00 | -0.00 – 0.00 | 0.302 | 0.00 | -0.00 – 0.00 | 0.450 |
|
Tract ID [right slft] × Time since injury |
-0.00 | -0.00 – 0.00 | 0.493 | 0.00 | -0.00 – 0.00 | 0.923 |
|
Tract ID [right unc] × Time since injury |
-0.00 | -0.00 – 0.00 | 0.307 | 0.00 | 0.00 – 0.00 | 0.043 |
| Time since injury^2 | 0.00 | 0.00 – 0.00 | 0.001 | |||
|
Tract ID [forceps minor] × Time since injury^2 |
-0.00 | -0.00 – 0.00 | 0.474 | |||
|
Tract ID [left atr] × Time since injury^2 |
-0.00 | -0.00 – 0.00 | 0.197 | |||
|
Tract ID [left cab] × Time since injury^2 |
-0.00 | -0.00 – -0.00 | 0.005 | |||
|
Tract ID [left cing] × Time since injury^2 |
-0.00 | -0.00 – 0.00 | 0.496 | |||
|
Tract ID [left cst] × Time since injury^2 |
-0.00 | -0.00 – 0.00 | 0.393 | |||
|
Tract ID [left ilf] × Time since injury^2 |
-0.00 | -0.00 – 0.00 | 0.699 | |||
|
Tract ID [left slfp] × Time since injury^2 |
-0.00 | -0.00 – 0.00 | 0.323 | |||
|
Tract ID [left slft] × Time since injury^2 |
-0.00 | -0.00 – 0.00 | 0.274 | |||
|
Tract ID [left unc] × Time since injury^2 |
-0.00 | -0.00 – 0.00 | 0.381 | |||
|
Tract ID [right atr] × Time since injury^2 |
-0.00 | -0.00 – 0.00 | 0.706 | |||
|
Tract ID [right cab] × Time since injury^2 |
-0.00 | -0.00 – -0.00 | 0.001 | |||
|
Tract ID [right cing] × Time since injury^2 |
-0.00 | -0.00 – 0.00 | 0.848 | |||
|
Tract ID [right cst] × Time since injury^2 |
-0.00 | -0.00 – 0.00 | 0.234 | |||
|
Tract ID [right ilf] × Time since injury^2 |
-0.00 | -0.00 – 0.00 | 0.211 | |||
|
Tract ID [right slfp] × Time since injury^2 |
-0.00 | -0.00 – 0.00 | 0.630 | |||
|
Tract ID [right slft] × Time since injury^2 |
-0.00 | -0.00 – 0.00 | 0.765 | |||
|
Tract ID [right unc] × Time since injury^2 |
-0.00 | -0.00 – -0.00 | 0.017 | |||
| Observations | 486 | 486 | ||||
| R2 / R2 adjusted | 0.993 / 0.993 | 0.995 / 0.994 | ||||
ANOVA COMPARISON FOR MODELS BY TRACT AND TIME
| Res.Df | RSS | Df | Sum of Sq | F | Pr(>F) |
|---|---|---|---|---|---|
| 450 | 0.01487 | ||||
| 432 | 0.01212 | 18 | 0.002743 | 5.431 | 1.614e-11 |
AIC FOR MODELS BY TRACT AND TIME
AIC for model 1 (i.e., linear model) -3599
AIC for model 2 (i.e.,quadratic model) -3662
INTERPRETATION
- For models by tract and time, quadratic models are preferred (see ANOVA results, AIC scores)
Mixed Models
Statistical Framework
Should Tract.ID be considered a control variable or a moderating variable?
- Control variables are measured but their effect is not hypothesized. Moderator variables are measured, and their effect is hypothesized.
- In the statistical models below, Tract.ID is hypothesized to directly affect FA, not the relationship between time since injury and FA. So, it can be considered a control variable.
- Reference: https://business.fiu.edu/academics/graduate/insights/posts/control-and-moderation-its-easy-when-you-understand.html#:~:text=You%20decide%20which%20is%20which,latter%2C%20it%20is%20a%20moderator.
Time-Dep-FA
Research Q: How does depression moderate the relationship between time since injury and FA trajectory/recovery?
Hypotheses:
- Time since injury has a positive effect on FA.
- The positive effect of FA on time since injury is stronger for individuals with lower depressive symptoms.
Proposed Linear Mixed Effects Model
time.Dep.FA.lme.model <- nlme::lme(zFA ~ zTime.since.injury*zDep_BDI,random=list(ParticipantID=~1, Tract.ID=~1), data=tbi)
Checking Model Assumptions
ASSUMPTION 1: Linearity
ASSUMPTION 2: Normal Distribution of Residuals
ASSUMPTION 3: Homoscedasticity of model and random effects
## OK: Error variance appears to be homoscedastic (p = 0.879).
ASSUMPTION 4: No Auto-Correlation of Residuals
ASSUMPTION 5: NO MULTI-COLLINEARITY
## # Check for Multicollinearity
##
## Low Correlation
##
## Term VIF VIF 95% CI Increased SE Tolerance
## zTime.since.injury 1.03 [1.00, 1.40] 1.01 0.97
## zDep_BDI 1.03 [1.00, 1.37] 1.02 0.97
## zTime.since.injury:zDep_BDI 1.01 [1.00, 13.12] 1.01 0.99
## Tolerance 95% CI
## [0.71, 1.00]
## [0.73, 1.00]
## [0.08, 1.00]Final Model Output
## Linear mixed-effects model fit by REML
## Data: tbi
## AIC BIC logLik
## -968.3397 -936.1501 491.1698
##
## Random effects:
## Formula: ~1 | ParticipantID
## (Intercept)
## StdDev: 0.0001090306
##
## Formula: ~1 | Tract.ID %in% ParticipantID
## (Intercept) Residual
## StdDev: 0.9638819 0.1019467
##
## Fixed effects: zFA ~ zTime.since.injury * zDep_BDI
## Value Std.Error DF t-value p-value
## (Intercept) -0.01741002 0.16070895 699 -0.108333 0.9138
## zTime.since.injury -0.03147903 0.00381327 699 -8.255130 0.0000
## zDep_BDI 0.03496452 0.00724110 699 4.828620 0.0000
## zTime.since.injury:zDep_BDI -0.00721407 0.00313233 699 -2.303096 0.0216
## Correlation:
## (Intr) zTm.s. zD_BDI
## zTime.since.injury 0.002
## zDep_BDI 0.013 0.156
## zTime.since.injury:zDep_BDI -0.001 -0.071 -0.083
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -4.20172246 -0.55503697 -0.01485419 0.55910632 4.49691635
##
## Number of Observations: 738
## Number of Groups:
## ParticipantID Tract.ID %in% ParticipantID
## 2 36## numDF denDF F-value p-value
## (Intercept) 1 699 0.02743 0.8685
## zTime.since.injury 1 699 85.56834 <.0001
## zDep_BDI 1 699 21.66266 <.0001
## zTime.since.injury:zDep_BDI 1 699 5.30425 0.0216Time-Anx-FA
Research Q: How does anxiety moderate the relationship between time since injury and FA trajectory/recovery?
Hypotheses:
- Time since injury has a positive effect on FA.
- The positive effect of FA on time since injury is stronger for individuals with lower anxiety symptoms.
Proposed Linear Mixed Effects Model
time.Anx.FA.lme.model <- nlme::lme(zFA ~ zTime.since.injury*zAnx_BAI,random=list(ParticipantID=~1, Tract.ID=~1), data=tbi)
Checking Model Assumptions
ASSUMPTION 1: Linearity
ASSUMPTION 2: Normal Distribution of Residuals
ASSUMPTION 3: Homoscedasticity of model and random effects
## OK: Error variance appears to be homoscedastic (p = 0.883).
ASSUMPTION 4: No Auto-Correlation of Residuals
ASSUMPTION 5: NO MULTI-COLLINEARITY
## # Check for Multicollinearity
##
## Low Correlation
##
## Term VIF VIF 95% CI Increased SE Tolerance
## zTime.since.injury 1.36 [1.25, 1.50] 1.16 0.74
## zAnx_BAI 1.95 [1.76, 2.18] 1.39 0.51
## zTime.since.injury:zAnx_BAI 1.51 [1.39, 1.68] 1.23 0.66
## Tolerance 95% CI
## [0.67, 0.80]
## [0.46, 0.57]
## [0.60, 0.72]Final Model Output
## Linear mixed-effects model fit by REML
## Data: tbi
## AIC BIC logLik
## -957.5886 -925.3991 485.7943
##
## Random effects:
## Formula: ~1 | ParticipantID
## (Intercept)
## StdDev: 0.0001030225
##
## Formula: ~1 | Tract.ID %in% ParticipantID
## (Intercept) Residual
## StdDev: 0.9652455 0.1027296
##
## Fixed effects: zFA ~ zTime.since.injury * zAnx_BAI
## Value Std.Error DF t-value p-value
## (Intercept) -0.025654550 0.16094147 699 -0.159403 0.8734
## zTime.since.injury -0.027729077 0.00441384 699 -6.282304 0.0000
## zAnx_BAI 0.015299363 0.00678076 699 2.256292 0.0244
## zTime.since.injury:zAnx_BAI -0.006003478 0.00419150 699 -1.432297 0.1525
## Correlation:
## (Intr) zTm.s. zA_BAI
## zTime.since.injury 0.006
## zAnx_BAI 0.013 0.491
## zTime.since.injury:zAnx_BAI 0.013 0.154 0.565
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -4.194119185 -0.561723620 -0.001038555 0.564403242 4.297677826
##
## Number of Observations: 738
## Number of Groups:
## ParticipantID Tract.ID %in% ParticipantID
## 2 36## numDF denDF F-value p-value
## (Intercept) 1 699 0.02735 0.8687
## zTime.since.injury 1 699 84.26891 <.0001
## zAnx_BAI 1 699 13.79525 0.0002
## zTime.since.injury:zAnx_BAI 1 699 2.05147 0.1525