Question A

Plot 1: Birth Weight, Age and Order

Plot 2: Weight and Maternal Age

Question B

  Model df      AIC      BIC    logLik   Test  L.Ratio p-value

gls.1 1 16 67062.39 67164.57 -33515.20
gls.2 2 7 67093.83 67138.53 -33539.92 1 vs 2 49.43719 <.0001 gls.3 3 7 67273.32 67318.02 -33629.66
[1] 1.840139e-11

The betas for order do not appear to differ by model correlation structure. The effect size estimates increase by increasing birth order (except for Order 4, which slightly decreases from Order 5). The models perform similarly. However, the AIC for the independent model is smaller than the other 2, and its log-likelihood is greater, which may indicate a better fit for the data.

Question C

**Random Intercept Model**

	Dependent variable:

	Wt

I(Age)	17.138^***
	(1.980)

Constant	2,785.254^***
	(44.993)


Observations	4,390
Log Likelihood	-33,530.540
Akaike Inf. Crit.	67,069.070
Bayesian Inf. Crit.	67,094.620

Note:	p<0.1; p<0.05; p<0.01

There appears to be evidence of an association between infant birth weight and maternal age. The fixed effects estimate for maternal age is 17.14 (p<0.001). This means that for every 1 year increase in maternal age, there is an estimated 17.14 gram increase in infant birth weight. For subject 247, the b_i is -147.02, which is it’s subject-specific intercept (as b_i is estimates the variability of the intercept across individuals). The conditional mean birthweight at age 20 is 2981.003, or 2981.0031.

Question D

**Fixed effects model**

	Dependent variable:

	Wt

I(Age)	11.832^***
	(2.343)


Observations	4,390
R²	0.985
Adjusted R²	0.982
Residual Std. Error	433.934 (df = 3511)
F Statistic	268.867^*** (df = 879; 3511)

Note:	p<0.1; p<0.05; p<0.01

There appears to be evidence of an association between birth weight and maternal age. The fixed effect estimate of maternal age is 11.83195 (p-value <0.01), which can be interpreted as the effect size of maternal age on birth weight after adjustment for subject-specific variation. In other words, after adjustment for confounders (or within-individual effects), 1 year increase in maternal age corresponds to an 11.83195 increase in infant birth weight (in grams).

Question E

**Random Intercept vs. Fixed Effects Models of Maternal Age**

	Dependent variable:

	Wt
	linear	OLS
	mixed effects
	Random Effects	Fixed Effects
	(1)	(2)

I(Age)	17.138^***	11.832^***
	(1.980)	(2.343)

Constant	2,785.254^***
	(44.993)


Observations	4,390	4,390
R²		0.985
Adjusted R²		0.982
Log Likelihood	-33,530.540
Akaike Inf. Crit.	67,069.070
Bayesian Inf. Crit.	67,094.620
Residual Std. Error		433.934 (df = 3511)
F Statistic		268.867^*** (df = 879; 3511)

Note:	p<0.1; p<0.05; p<0.01

In the random intercept model, the effect size for maternal age is 17.14 (p<0.001), and in the fixed effects model, the effect size for maternal age is 11.83195 (p-value <0.01). While the prior model takes into account the variability of the intercept across individuals (or the variance that can be attributed to differences between individuals), the fixed effects model adjusts for all other unmeasured confounding of the indivudals. This could explain the smaller effect size of maternal age in the fixed effects model, if it is the case that the \(\hat{\beta_1}^{(RE)}\) was inflated due to unmeasured confounding.

Question F

## Linear mixed-effects model fit by REML
##  Data: birthwt.means 
##       AIC      BIC    logLik
##   67048.3 67080.24 -33519.15
## 
## Random effects:
##  Formula: ~1 | MID
##         (Intercept) Residual
## StdDev:    351.8482 433.9342
## 
## Fixed effects: Wt ~ 1 + I(mageMID) + I(Age - mageMID) 
##                     Value Std.Error   DF   t-value p-value
## (Intercept)      2499.108  80.69220 3511 30.970879       0
## I(mageMID)         30.355   3.67406  876  8.261983       0
## I(Age - mageMID)   11.832   2.34256 3511  5.050863       0
##  Correlation: 
##                  (Intr) I(MID)
## I(mageMID)       -0.986       
## I(Age - mageMID)  0.000  0.000
## 
## Standardized Within-Group Residuals:
##         Min          Q1         Med          Q3         Max 
## -6.00869342 -0.46679435  0.05271805  0.56461728  3.15143540 
## 
## Number of Observations: 4390
## Number of Groups: 878

The fixed effects estimate for the cross-sectional effect of maternal age is 30.35 (p < 0.001). The \(\hat{\beta_1}^{(C)}\) can be interpreted as the effect size of individual’s mean age on infant birth weight. The fixed effects estimate for the longitudinal effect of maternal age is 11.83 (p < 0.001). The \(\hat{\beta_1}^{(L)}\) can be interpreted as the effect size of a subject’s deviation from their mean maternal age on infant birth weight.

For example, Individual A who had live births at ages 20, 30, and 40 will have the same \(\hat{\beta_1}^{(C)}\) as Individual B who had live births at ages 29, 30, and 31 (as mean maternal age for both is 30). In addition, the \(\hat{\beta_1}^{(C)}\) will be the same for each \(n^{th}\) birth of an subject. \(\hat{\beta_1}^{(L)}\), however, will take into account the deviation, or general dispersion, from mean individual maternal age. Therefore, \(\hat{\beta_1}^{(L)}\) will be different Individual A and Individual B, and will take into maternal age dispersion of their respective live births.

Question G

## Linear mixed-effects model fit by REML
##  Data: birthwt 
##        AIC      BIC    logLik
##   67013.36 67051.68 -33500.68
## 
## Random effects:
##  Formula: ~1 + Order | MID
##  Structure: Diagonal
##         (Intercept)  Order Residual
## StdDev:    307.4832 60.289 423.1695
## 
## Fixed effects: Wt ~ 1 + I(Age) + Order 
##                 Value Std.Error   DF  t-value p-value
## (Intercept) 2669.3128  56.32034 3510 47.39518  0.0000
## I(Age)        26.0611   3.32030 3510  7.84904  0.0000
## Order        -25.7471   7.93425 3510 -3.24505  0.0012
##  Correlation: 
##        (Intr) I(Age)
## I(Age) -0.946       
## Order   0.602 -0.781
## 
## Standardized Within-Group Residuals:
##         Min          Q1         Med          Q3         Max 
## -5.43285638 -0.45682762  0.05688877  0.56140966  3.33450884 
## 
## Number of Observations: 4390
## Number of Groups: 878

This model adds a varying slope of birth order by MID. The standard deviation of random effect of birth order is 60.3. I would interpret this as the deviation from the mean slope and intercept (or deviation from fixed effects?). As the standard deviation (or variance) of the random slope is not 0, and the AIC is slightly smaller with the random slope and intercept model compared to the random slope model, I would assume that this effect may provide some (small) improvement in the model.

HW1 - Val Pocus

Val Pocus