Per Nikki, participant 8194162701 was excluded.
75 participants were selected based on the availability of plasma samples and matches by race and sex, but one participant was excluded. The design called for 25 participants in 3 age groups: young (30-35), middle aged (45-50) and old (60-65). Despite our best intentions, discrepancies between the published inventory and the actual contents of the bank yielded the following wave 1 age distribution of sex and race matched participants after excluding one participant:
Age0grp
30-34 35-39 40-44 45-49 50-54 55-59 60-64
58 2 4 52 4 4 24
Classifying wave 1 age into 3 broad categories and descriptive statistics for age within each broad category:
ageGroups
Young Middle Old
60 60 28
ageGroups Age0.n Age0.mean Age0.sd Age0.min Age0.max
1 Young 60 31.86666667 1.371213327 30 35
2 Middle 60 46.90000000 2.621294802 40 53
3 Old 28 60.64285714 2.422338719 55 64
The overall interval between repeated assessments was 4.6 years and the intervals are roughly equivalent by broad age categories.
ageGroups ageDiff.n ageDiff.mean ageDiff.sd ageDiff.min ageDiff.max
1 Young 30 4.671686061 1.1281334557 2.12457221 7.34291581
2 Middle 30 4.600319416 1.0781281324 2.22313484 7.35112936
3 Old 14 4.537987681 0.7327750257 2.40383299 5.46475017
Including repeated assessments and matching requires us to account for the tendency of participants to resemble themselves over time and the likelihood that those who were matched will resemble one another more than they resemble others. In practical terms, this means that we can't use ordinary regression without accounting for the potential biases of these 'built-in' correlations. Additionally, although there are 150 assays, the degrees of freedom for analyses must start with the number of independent participants. In this design, there are 25 independent assessments, each of which consists of 3 participants matched by sex and race with 2 repeated assays.
Despite these complexities, we will perform a regression-like analysis (mixed-model regression, sometimes called random-effects regression). The results from these analyses are similar to those of other regression techniques in that we will examine the relationship between predictors and covariates on the outcome (concentration of extracellular vesicles).
Here's the distribution of concentrations (×10-9 ) by age (ignoring matching) for times 1 and 2.
Various colored dots in the scatterplot indicate BMIs.
ageGroups reConc.n reConc.mean reConc.sd reConc.min reConc.max
1 Young 30 132.66333333 100.99363409 16.80 423
2 Middle 30 97.79233333 89.27647007 4.56 353
3 Old 14 82.31000000 56.05350891 8.04 230
ageGroups reConc.n reConc.mean reConc.sd reConc.min reConc.max
1 Young 30 143.12866667 114.55783920 7.53 593
2 Middle 30 101.14933333 95.79644792 6.06 393
3 Old 14 70.06928571 60.87163524 3.07 184
Regression techniques assume that mean levels and variances are independent. The scatterplots suggest that we may violate that assumption becasue the spread of concentrations at younger ages is greater than the spread of concentrations at 'middle' ages, which in turn is greater than the spread of concerntrations at older ages. Sometimes transformations can ameliorate the violation. However, it isn't clear that transformations will help these data, not the least because the middle age group mean is predictable from the relationship between the mean and the variance.
For the moment, we can ignore the repeated measures and examine whether concentrations are associated with age differences. We can take advantage of the repeated measures by using the 2nd assessment as a replication. However, we still need to account for matching in these data so the appropriate technique is still mixed models.
We will examine age differences and the contributions of BMI, cigarette smoking (current v. not), and diabetes diagnoses in analyses that follow this pattern for BMI: with age, with age, BMI, and age×BMI, and with age and BMI. For numerical convenience we use cenAge (centered age) which is (Age - 50) / 10 and is interpreted as the effect of age decade. Note that t-values >|1.96| are significant at p<.05. The analysis of variance table following each analysis displays the exact p-value based on a F test (with 1 df F = t2 ).
Linear mixed model fit by REML
t-tests use Satterthwaite approximations to degrees of freedom ['lmerMod']
Formula: reConc ~ cenAge + (1 | Seq)
Data: subset(exConc, HNDwave == 1)
REML criterion at convergence: 859.2
Scaled residuals:
Min 1Q Median 3Q Max
-1.3791332 -0.6575433 -0.1684924 0.2435731 3.1631500
Random effects:
Groups Name Variance Std.Dev.
Seq (Intercept) 805.6609 28.38417
Residual 7150.3944 84.56000
Number of obs: 74, groups: Seq, 31
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 95.955305 12.551148 36.727390 7.64514 0.0000000041834
cenAge -18.937614 9.084368 51.883780 -2.08464 0.042042
Correlation of Fixed Effects:
(Intr)
cenAge 0.458
Analysis of variance
Analysis of Variance Table of type III with Satterthwaite
approximation for degrees of freedom
Sum Sq Mean Sq NumDF DenDF F.value Pr(>F)
cenAge 31073.564 31073.564 1 51.883775 4.3457133 0.042042
Linear mixed model fit by REML
t-tests use Satterthwaite approximations to degrees of freedom ['lmerMod']
Formula: reConc ~ cenAge * BMI + (1 | Seq)
Data: subset(exConc, HNDwave == 1)
REML criterion at convergence: 850.3
Scaled residuals:
Min 1Q Median 3Q Max
-1.2866679 -0.5585555 -0.2168830 0.3632742 3.2977562
Random effects:
Groups Name Variance Std.Dev.
Seq (Intercept) 730.0089 27.01868
Residual 7164.3948 84.64275
Number of obs: 74, groups: Seq, 31
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) -1.793760 64.560959 65.141980 -0.02778 0.97792
cenAge -49.731147 53.145526 68.204890 -0.93575 0.35270
BMI 3.632485 2.336937 64.497420 1.55438 0.12499
cenAge:BMI 1.063406 1.919061 68.623520 0.55413 0.58129
Correlation of Fixed Effects:
(Intr) cenAge BMI
cenAge 0.530
BMI -0.981 -0.497
cenAge:BMI -0.502 -0.985 0.483
Analysis of variance
Analysis of Variance Table of type III with Satterthwaite
approximation for degrees of freedom
Sum Sq Mean Sq NumDF DenDF F.value Pr(>F)
cenAge 6273.4009 6273.4009 1 68.204888 0.87563584 0.35270
BMI 17309.8529 17309.8529 1 64.497419 2.41609421 0.12499
cenAge:BMI 2199.8873 2199.8873 1 68.623517 0.30705836 0.58129
Linear mixed model fit by REML
t-tests use Satterthwaite approximations to degrees of freedom ['lmerMod']
Formula: reConc ~ cenAge + BMI + (1 | Seq)
Data: subset(exConc, HNDwave == 1)
REML criterion at convergence: 853.7
Scaled residuals:
Min 1Q Median 3Q Max
-1.3842105 -0.5739972 -0.2250633 0.3385274 3.4392234
Random effects:
Groups Name Variance Std.Dev.
Seq (Intercept) 699.648 26.45086
Residual 7114.370 84.34673
Number of obs: 74, groups: Seq, 31
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 16.142355 55.590446 67.678410 0.29038 0.772413
cenAge -20.683763 9.142953 50.191910 -2.26226 0.028042
BMI 3.009890 2.036966 66.881270 1.47763 0.144200
Correlation of Fixed Effects:
(Intr) cenAge
cenAge 0.241
BMI -0.975 -0.143
Analysis of variance
Analysis of Variance Table of type III with Satterthwaite
approximation for degrees of freedom
Sum Sq Mean Sq NumDF DenDF F.value Pr(>F)
cenAge 36410.156 36410.156 1 50.191910 5.1178327 0.028042
BMI 15533.526 15533.526 1 66.881273 2.1834014 0.144200
These results suggest that there are significant age differences in concentrations, but no association of BMI with concentrations after adjusting for age differences.
Linear mixed model fit by REML
t-tests use Satterthwaite approximations to degrees of freedom ['lmerMod']
Formula: reConc ~ cenAge + (1 | Seq)
Data: subset(exConc, HNDwave == 3)
REML criterion at convergence: 873.3
Scaled residuals:
Min 1Q Median 3Q Max
-1.3337136 -0.6127577 -0.2133045 0.3330396 4.5691852
Random effects:
Groups Name Variance Std.Dev.
Seq (Intercept) 0.000 0.00000
Residual 9593.666 97.94726
Number of obs: 74, groups: Seq, 31
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 108.28869 11.49545 72.01172 9.42014 3.4639e-14
cenAge -26.38639 10.43440 72.01172 -2.52879 0.013637
Correlation of Fixed Effects:
(Intr)
cenAge 0.138
Analysis of variance
Analysis of Variance Table of type III with Satterthwaite
approximation for degrees of freedom
Sum Sq Mean Sq NumDF DenDF F.value Pr(>F)
cenAge 61349.322 61349.322 1 72.011722 6.3947735 0.013637
Linear mixed model fit by REML
t-tests use Satterthwaite approximations to degrees of freedom ['lmerMod']
Formula: reConc ~ cenAge * BMI + (1 | Seq)
Data: subset(exConc, HNDwave == 3)
REML criterion at convergence: 862.5
Scaled residuals:
Min 1Q Median 3Q Max
-1.5755270 -0.6344126 -0.1406293 0.3473013 4.3535641
Random effects:
Groups Name Variance Std.Dev.
Seq (Intercept) 0.000 0.00000
Residual 9290.698 96.38827
Number of obs: 74, groups: Seq, 31
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) -8.8733720 59.0239946 69.9488400 -0.15033 0.880933
cenAge -38.6296558 55.8015070 69.9488400 -0.69227 0.491060
BMI 4.3561974 2.1554610 69.9488400 2.02100 0.047108
cenAge:BMI 0.4698818 1.9726460 69.9488400 0.23820 0.812423
Correlation of Fixed Effects:
(Intr) cenAge BMI
cenAge 0.338
BMI -0.981 -0.343
cenAge:BMI -0.346 -0.983 0.355
Analysis of variance
Analysis of Variance Table of type III with Satterthwaite
approximation for degrees of freedom
Sum Sq Mean Sq NumDF DenDF F.value Pr(>F)
cenAge 4452.441 4452.441 1 69.948836 0.4792364 0.491060
BMI 37947.494 37947.494 1 69.948836 4.0844611 0.047108
cenAge:BMI 527.142 527.142 1 69.948836 0.0567386 0.812423
Linear mixed model fit by REML
t-tests use Satterthwaite approximations to degrees of freedom ['lmerMod']
Formula: reConc ~ cenAge + BMI + (1 | Seq)
Data: subset(exConc, HNDwave == 3)
REML criterion at convergence: 865.8
Scaled residuals:
Min 1Q Median 3Q Max
-1.6025224 -0.6315373 -0.1148764 0.3229290 4.3508946
Random effects:
Groups Name Variance Std.Dev.
Seq (Intercept) 0.000 0.00000
Residual 9167.268 95.74585
Number of obs: 74, groups: Seq, 31
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) -4.008576 55.008905 70.973660 -0.07287 0.942113
cenAge -25.565128 10.207481 70.973660 -2.50455 0.014560
BMI 4.173796 2.001424 70.973660 2.08541 0.040629
Correlation of Fixed Effects:
(Intr) cenAge
cenAge -0.010
BMI -0.979 0.039
Analysis of variance
Analysis of Variance Table of type III with Satterthwaite
approximation for degrees of freedom
Sum Sq Mean Sq NumDF DenDF F.value Pr(>F)
cenAge 57504.084 57504.084 1 70.973661 6.2727616 0.014560
BMI 39867.967 39867.967 1 70.973661 4.3489477 0.040629
These results suggest that there are significant age differences in concentrations, and significant associations of BMI with concentrations after adjusting for age differences, but no interaction between age×BMI.
The effects of age and BMI are not consistent. Although there age differences in concentrations at waves 1 and 3, BMI was associated with concentration only at wave 3. There was no evidence for an age×BMI interaction in either wave.
Linear mixed model fit by REML
t-tests use Satterthwaite approximations to degrees of freedom ['lmerMod']
Formula: reConc ~ cenAge * CigaretteCurr + (1 | Seq)
Data: subset(exConc, HNDwave == 1)
REML criterion at convergence: 757.8
Scaled residuals:
Min 1Q Median 3Q Max
-1.4507352 -0.6668698 -0.1948412 0.2699308 3.0014905
Random effects:
Groups Name Variance Std.Dev.
Seq (Intercept) 461.9773 21.49366
Residual 7389.1846 85.96037
Number of obs: 67, groups: Seq, 31
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 100.86084 21.30177 62.59013 4.73486 0.00001298
cenAge -26.31923 18.38601 56.56912 -1.43148 0.15779
CigaretteCurrNo -11.34838 25.86542 59.05578 -0.43875 0.66245
cenAge:CigaretteCurrNo 12.08707 21.79749 62.15153 0.55452 0.58122
Correlation of Fixed Effects:
(Intr) cenAge CgrtCN
cenAge 0.611
CigarttCrrN -0.804 -0.509
cnAg:CgrtCN -0.517 -0.853 0.536
Analysis of variance
Analysis of Variance Table of type III with Satterthwaite
approximation for degrees of freedom
Sum Sq Mean Sq NumDF DenDF F.value Pr(>F)
cenAge 26407.8052 26407.8052 1 46.756012 3.5738456 0.064903
CigaretteCurr 1422.4106 1422.4106 1 59.055637 0.1924990 0.662446
cenAge:CigaretteCurr 2272.0896 2272.0896 1 62.151467 0.3074885 0.581215
Linear mixed model fit by REML
t-tests use Satterthwaite approximations to degrees of freedom ['lmerMod']
Formula: reConc ~ cenAge + CigaretteCurr + (1 | Seq)
Data: subset(exConc, HNDwave == 1)
REML criterion at convergence: 766.1
Scaled residuals:
Min 1Q Median 3Q Max
-1.4281275 -0.6541020 -0.2288228 0.3130376 3.1313021
Random effects:
Groups Name Variance Std.Dev.
Seq (Intercept) 526.8148 22.95245
Residual 7245.2637 85.11911
Number of obs: 67, groups: Seq, 31
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 106.815336 18.153964 60.460100 5.88386 0.00000018774
cenAge -17.751902 9.519552 46.318220 -1.86478 0.068555
CigaretteCurrNo -19.025835 21.675690 56.865450 -0.87775 0.383773
Correlation of Fixed Effects:
(Intr) cenAge
cenAge 0.379
CigarttCrrN -0.726 -0.117
Analysis of variance
Analysis of Variance Table of type III with Satterthwaite
approximation for degrees of freedom
Sum Sq Mean Sq NumDF DenDF F.value Pr(>F)
cenAge 25194.8035 25194.8035 1 46.318221 3.477417 0.068555
CigaretteCurr 5582.0769 5582.0769 1 56.865452 0.770445 0.383773
Linear mixed model fit by REML
t-tests use Satterthwaite approximations to degrees of freedom ['lmerMod']
Formula: reConc ~ cenAge * CigaretteCurr + (1 | Seq)
Data: subset(exConc, HNDwave == 3)
REML criterion at convergence: 823
Scaled residuals:
Min 1Q Median 3Q Max
-1.4063708 -0.6729284 -0.1416325 0.3388017 4.0324733
Random effects:
Groups Name Variance Std.Dev.
Seq (Intercept) 0.000 0.00000
Residual 8575.689 92.60502
Number of obs: 72, groups: Seq, 31
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 150.11471 20.48096 68.00227 7.32948 0.00000000036053
cenAge -49.30434 20.63773 68.00227 -2.38904 0.019673
CigaretteCurrNo -61.81026 24.38694 68.00227 -2.53456 0.013566
cenAge:CigaretteCurrNo 34.15838 23.60240 68.00227 1.44724 0.152424
Correlation of Fixed Effects:
(Intr) cenAge CgrtCN
cenAge 0.333
CigarttCrrN -0.840 -0.280
cnAg:CgrtCN -0.291 -0.874 0.255
Analysis of variance
Analysis of Variance Table of type III with Satterthwaite
approximation for degrees of freedom
Sum Sq Mean Sq NumDF DenDF F.value Pr(>F)
cenAge 63944.982 63944.982 1 68.026646 7.4565416 0.0080419
CigaretteCurr 55090.340 55090.340 1 68.026646 6.4240132 0.0135648
cenAge:CigaretteCurr 17961.853 17961.853 1 68.026646 2.0945085 0.1524220
Linear mixed model fit by REML
t-tests use Satterthwaite approximations to degrees of freedom ['lmerMod']
Formula: reConc ~ cenAge + CigaretteCurr + (1 | Seq)
Data: subset(exConc, HNDwave == 3)
REML criterion at convergence: 833.3
Scaled residuals:
Min 1Q Median 3Q Max
-1.5532204 -0.6217115 -0.1730476 0.4046762 4.3024800
Random effects:
Groups Name Variance Std.Dev.
Seq (Intercept) 1.942983e-12 0.000001393909
Residual 8.711720e+03 93.336597531700
Number of obs: 72, groups: Seq, 31
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 158.75449 19.74637 69.00667 8.03968 0.000000000016903
cenAge -23.18823 10.09299 69.00667 -2.29746 0.0246303
CigaretteCurrNo -70.79369 23.77005 69.00667 -2.97827 0.0039975
Correlation of Fixed Effects:
(Intr) cenAge
cenAge 0.169
CigarttCrrN -0.828 -0.122
Analysis of variance
Analysis of Variance Table of type III with Satterthwaite
approximation for degrees of freedom
Sum Sq Mean Sq NumDF DenDF F.value Pr(>F)
cenAge 45983.197 45983.197 1 69.006674 5.2783141 0.0246303
CigaretteCurr 77273.873 77273.873 1 69.006674 8.8701047 0.0039975
TBA
Linear mixed model fit by REML
t-tests use Satterthwaite approximations to degrees of freedom ['lmerMod']
Formula: reConc ~ cenAge * dxDiabetes + (1 | Seq)
Data: subset(exConc, HNDwave == 1)
REML criterion at convergence: 818.3
Scaled residuals:
Min 1Q Median 3Q Max
-1.5086151 -0.6019720 -0.1430443 0.2107774 3.2062691
Random effects:
Groups Name Variance Std.Dev.
Seq (Intercept) 893.2596 29.88745
Residual 7060.3499 84.02589
Number of obs: 74, groups: Seq, 31
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 92.192046 15.209003 51.183640 6.06168 0.00000016221
cenAge -17.833665 10.615384 53.342170 -1.67998 0.098809
dxDiabetespreDiabetes -2.157412 35.778705 62.456370 -0.06030 0.952110
dxDiabetesDiabetes 50.613133 43.004717 65.904130 1.17692 0.243460
cenAge:dxDiabetespreDiabetes 6.566323 43.777806 66.504480 0.14999 0.881225
cenAge:dxDiabetesDiabetes -52.169193 41.004365 67.121620 -1.27228 0.207662
Correlation of Fixed Effects:
(Intr) cenAge dxDbtspD dxDbtsDb cnAg:dxDbtspD
cenAge 0.587
dxDbtsprDbt -0.378 -0.245
dxDibtsDbts -0.329 -0.220 0.152
cnAg:dxDbtspD -0.140 -0.253 -0.422 0.035
cnAg:dxDbtsDb -0.162 -0.285 0.047 0.265 0.087
Analysis of variance
Analysis of Variance Table of type III with Satterthwaite
approximation for degrees of freedom
Sum Sq Mean Sq NumDF DenDF F.value Pr(>F)
cenAge 19955.406 19955.4059 1 66.870201 2.82640466 0.097391
dxDiabetes 10194.282 5097.1411 2 63.761455 0.72193888 0.489737
cenAge:dxDiabetes 11913.712 5956.8558 2 65.584405 0.84370547 0.434728
Linear mixed model fit by REML
t-tests use Satterthwaite approximations to degrees of freedom ['lmerMod']
Formula: reConc ~ cenAge + dxDiabetes + (1 | Seq)
Data: subset(exConc, HNDwave == 1)
REML criterion at convergence: 838.6
Scaled residuals:
Min 1Q Median 3Q Max
-1.5787402 -0.5883902 -0.1703749 0.2715641 3.2557299
Random effects:
Groups Name Variance Std.Dev.
Seq (Intercept) 752.656 27.43458
Residual 7142.912 84.51575
Number of obs: 74, groups: Seq, 31
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 90.191208 14.761897 49.824950 6.10973 0.00000014958
cenAge -20.642236 9.913014 51.633720 -2.08234 0.042284
dxDiabetespreDiabetes 2.921658 32.322502 62.678750 0.09039 0.928265
dxDiabetesDiabetes 64.938211 41.433454 68.158020 1.56729 0.121678
Correlation of Fixed Effects:
(Intr) cenAge dxDbtspD
cenAge 0.566
dxDbtsprDbt -0.485 -0.392
dxDibtsDbts -0.303 -0.156 0.165
Analysis of variance
Analysis of Variance Table of type III with Satterthwaite
approximation for degrees of freedom
Sum Sq Mean Sq NumDF DenDF F.value Pr(>F)
cenAge 30972.58 30972.580 1 51.659207 4.3361281 0.042282
dxDiabetes 17752.07 8876.035 2 65.261260 1.2426354 0.295361
Linear mixed model fit by REML
t-tests use Satterthwaite approximations to degrees of freedom ['lmerMod']
Formula: reConc ~ cenAge * dxDiabetes + (1 | Seq)
Data: subset(exConc, HNDwave == 3)
REML criterion at convergence: 824.7
Scaled residuals:
Min 1Q Median 3Q Max
-1.8802025 -0.6416521 -0.1485437 0.4333240 3.2611365
Random effects:
Groups Name Variance Std.Dev.
Seq (Intercept) 0.000 0.00000
Residual 8434.723 91.84075
Number of obs: 74, groups: Seq, 31
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 93.74380 12.85210 68.01119 7.29405 0.00000000041759
cenAge -21.40072 11.85212 68.01119 -1.80565 0.075399
dxDiabetespreDiabetes 78.29328 29.67046 68.01119 2.63876 0.010307
dxDiabetesDiabetes 90.78288 46.27386 68.01119 1.96186 0.053873
cenAge:dxDiabetespreDiabetes -68.12270 28.20743 68.01119 -2.41506 0.018426
cenAge:dxDiabetesDiabetes -21.57356 33.92769 68.01119 -0.63587 0.526996
Correlation of Fixed Effects:
(Intr) cenAge dxDbtspD dxDbtsDb cnAg:dxDbtspD
cenAge 0.297
dxDbtsprDbt -0.433 -0.129
dxDibtsDbts -0.278 -0.082 0.120
cnAg:dxDbtspD -0.125 -0.420 -0.195 0.035
cnAg:dxDbtsDb -0.104 -0.349 0.045 -0.316 0.147
Analysis of variance
Analysis of Variance Table of type III with Satterthwaite
approximation for degrees of freedom
Sum Sq Mean Sq NumDF DenDF F.value Pr(>F)
cenAge 110600.059 110600.059 1 68.03516 13.1124708 0.00056002
dxDiabetes 81874.455 40937.228 2 68.03516 4.8534170 0.01070594
cenAge:dxDiabetes 49878.260 24939.130 2 68.03516 2.9567219 0.05870571
Linear mixed model fit by REML
t-tests use Satterthwaite approximations to degrees of freedom ['lmerMod']
Formula: reConc ~ cenAge + dxDiabetes + (1 | Seq)
Data: subset(exConc, HNDwave == 3)
REML criterion at convergence: 847.9
Scaled residuals:
Min 1Q Median 3Q Max
-1.4633302 -0.5661051 -0.1676745 0.3287473 4.1328231
Random effects:
Groups Name Variance Std.Dev.
Seq (Intercept) 5.192685e-11 0.000007206029
Residual 8.906277e+03 94.373075876343
Number of obs: 74, groups: Seq, 31
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 89.55661 13.05364 70.01621 6.86066 0.0000000022373
cenAge -34.40816 10.46877 70.01621 -3.28674 0.0015863
dxDiabetespreDiabetes 64.94329 29.81682 70.01621 2.17808 0.0327706
dxDiabetesDiabetes 90.38811 44.95276 70.01621 2.01074 0.0482059
Correlation of Fixed Effects:
(Intr) cenAge dxDbtspD
cenAge 0.258
dxDbtsprDbt -0.467 -0.225
dxDibtsDbts -0.323 -0.200 0.163
Analysis of variance
Analysis of Variance Table of type III with Satterthwaite
approximation for degrees of freedom
Sum Sq Mean Sq NumDF DenDF F.value Pr(>F)
cenAge 96211.579 96211.579 1 69.990896 10.8026703 0.0015865
dxDiabetes 67304.541 33652.271 2 69.990896 3.7784889 0.0276510
TBA
In longitudinal analyses, the age term in the mixed model is interpreted as a measure of time, not age differences. A significant 'age' effect is therefore evidence for significant change over time, and a significant interaction with 'age' is evidence for differences in rates of change over time.
Linear mixed model fit by REML
t-tests use Satterthwaite approximations to degrees of freedom ['lmerMod']
Formula: reConc ~ cenAge + (0 + cenAge | HNDid) + (1 | Seq)
Data: exConc
Control: lmerControl(check.nobs.vs.nRE = "warning")
REML criterion at convergence: 1727
Scaled residuals:
Min 1Q Median 3Q Max
-2.2967027 -0.4640868 -0.1333725 0.2596862 4.8820951
Random effects:
Groups Name Variance Std.Dev.
HNDid cenAge 2296.213 47.91882
Seq (Intercept) 2241.810 47.34775
Residual 4294.351 65.53130
Number of obs: 148, groups: HNDid, 74; Seq, 31
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 98.145113 11.396547 26.421380 8.61183 0.0000000037835
cenAge -21.622809 8.634014 39.476290 -2.50437 0.016503
Correlation of Fixed Effects:
(Intr)
cenAge 0.158
Analysis of variance
Analysis of Variance Table of type III with Satterthwaite
approximation for degrees of freedom
Sum Sq Mean Sq NumDF DenDF F.value Pr(>F)
cenAge 26933.714 26933.714 1 39.476291 6.2718937 0.016503
Linear mixed model fit by REML
t-tests use Satterthwaite approximations to degrees of freedom ['lmerMod']
Formula: reConc ~ cenAge * ageGroups + (0 + cenAge | HNDid) + (1 | Seq)
Data: exConc
Control: lmerControl(check.nobs.vs.nRE = "warning")
REML criterion at convergence: 1687.9
Scaled residuals:
Min 1Q Median 3Q Max
-2.2425650 -0.4095063 -0.1251300 0.2239026 4.8297478
Random effects:
Groups Name Variance Std.Dev.
HNDid cenAge 2277.816 47.72648
Seq (Intercept) 2296.529 47.92212
Residual 4358.348 66.01778
Number of obs: 148, groups: HNDid, 74; Seq, 31
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 112.775977 52.996403 73.762170 2.12799 0.036679
cenAge -13.959025 34.850492 84.632120 -0.40054 0.689768
ageGroupsMiddle -12.477770 53.275549 67.732990 -0.23421 0.815527
ageGroupsOld -42.785919 81.715944 79.322590 -0.52359 0.602021
cenAge:ageGroupsMiddle 31.852705 45.920779 98.752560 0.69364 0.489533
cenAge:ageGroupsOld 4.449775 59.032563 103.229400 0.07538 0.940060
Correlation of Fixed Effects:
(Intr) cenAge agGrpM agGrpO cnA:GM
cenAge 0.942
ageGrpsMddl -0.972 -0.941
ageGropsOld -0.640 -0.619 0.640
cnAg:gGrpsM -0.717 -0.761 0.728 0.484
cnAg:gGrpsO -0.551 -0.586 0.550 -0.216 0.432
Analysis of variance
Analysis of Variance Table of type III with Satterthwaite
approximation for degrees of freedom
Sum Sq Mean Sq NumDF DenDF F.value Pr(>F)
cenAge 31.26299 31.26299 1 105.049687 0.007173128 0.93267
ageGroups 1270.42659 635.21330 2 77.326024 0.145746354 0.86461
cenAge:ageGroups 2367.44747 1183.72374 2 105.298183 0.271599194 0.76269
Linear mixed model fit by REML
t-tests use Satterthwaite approximations to degrees of freedom ['lmerMod']
Formula: reConc ~ cenAge * BMI + (0 + cenAge | HNDid) + (1 | Seq)
Data: exConc
Control: lmerControl(check.nobs.vs.nRE = "warning")
REML criterion at convergence: 1714.7
Scaled residuals:
Min 1Q Median 3Q Max
-2.2794351 -0.4862379 -0.1116986 0.2789239 4.5650927
Random effects:
Groups Name Variance Std.Dev.
HNDid cenAge 2392.520 48.91339
Seq (Intercept) 1811.868 42.56605
Residual 4186.528 64.70338
Number of obs: 148, groups: HNDid, 74; Seq, 31
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) -2.5011849 42.0557203 130.8765000 -0.05947 0.952666
cenAge -16.4682427 45.3091400 73.8905400 -0.36346 0.717296
BMI 3.8839059 1.5545634 133.9578700 2.49839 0.013686
cenAge:BMI -0.1919826 1.6443767 72.5344300 -0.11675 0.907380
Correlation of Fixed Effects:
(Intr) cenAge BMI
cenAge 0.318
BMI -0.967 -0.312
cenAge:BMI -0.312 -0.981 0.313
Analysis of variance
Analysis of Variance Table of type III with Satterthwaite
approximation for degrees of freedom
Sum Sq Mean Sq NumDF DenDF F.value Pr(>F)
cenAge 553.0658 553.0658 1 73.890542 0.1321061 0.717296
BMI 26132.1101 26132.1101 1 133.957870 6.2419534 0.013686
cenAge:BMI 57.0657 57.0657 1 72.534433 0.0136308 0.907380
Linear mixed model fit by REML
t-tests use Satterthwaite approximations to degrees of freedom ['lmerMod']
Formula: reConc ~ cenAge + BMI + (0 + cenAge | HNDid) + (1 | Seq)
Data: exConc
Control: lmerControl(check.nobs.vs.nRE = "warning")
REML criterion at convergence: 1717.5
Scaled residuals:
Min 1Q Median 3Q Max
-2.2590753 -0.4952630 -0.1042330 0.2763242 4.5876287
Random effects:
Groups Name Variance Std.Dev.
HNDid cenAge 2305.072 48.01116
Seq (Intercept) 1787.044 42.27344
Residual 4199.589 64.80424
Number of obs: 148, groups: HNDid, 74; Seq, 31
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) -3.849555 39.784411 131.295650 -0.09676 0.9230642
cenAge -21.725121 8.588779 39.348750 -2.52948 0.0155376
BMI 3.930299 1.469756 134.115650 2.67412 0.0084244
Correlation of Fixed Effects:
(Intr) cenAge
cenAge 0.066
BMI -0.963 -0.023
Analysis of variance
Analysis of Variance Table of type III with Satterthwaite
approximation for degrees of freedom
Sum Sq Mean Sq NumDF DenDF F.value Pr(>F)
cenAge 26870.042 26870.042 1 39.348746 6.3982552 0.0155376
BMI 30030.837 30030.837 1 134.115646 7.1508991 0.0084244
From these analyses, concentrations decline over time but are associated directly with BMI such that concentrations are greater at greater BMIs. However, the absence of an age×BMI interaction suggests that change over time is unrelated to the association of BMI with concentration. These results are illustrated in the plot showing the predicting change in concentration in 3 BMI groups (22.5, normal; 27.5, overweight; and 32.5, obese).
Linear mixed model fit by REML
t-tests use Satterthwaite approximations to degrees of freedom ['lmerMod']
Formula: reConc ~ cenAge * CigaretteCurr + (0 + cenAge | HNDid) + (1 | Seq)
Data: exConc
Control: lmerControl(check.nobs.vs.nRE = "warning")
REML criterion at convergence: 1605.2
Scaled residuals:
Min 1Q Median 3Q Max
-1.9475190 -0.4710126 -0.1359470 0.2661056 4.6179326
Random effects:
Groups Name Variance Std.Dev.
HNDid cenAge 1784.920 42.24832
Seq (Intercept) 1403.093 37.45788
Residual 4957.886 70.41226
Number of obs: 139, groups: HNDid, 73; Seq, 31
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 116.338017 14.971074 74.698840 7.77085 0.000000000033278
cenAge -21.579008 16.008156 56.093040 -1.34800 0.183077
CigaretteCurrNo -29.153713 17.190076 134.635990 -1.69596 0.092204
cenAge:CigaretteCurrNo 3.474321 19.035273 57.917920 0.18252 0.855812
Correlation of Fixed Effects:
(Intr) cenAge CgrtCN
cenAge 0.366
CigarttCrrN -0.717 -0.337
cnAg:CgrtCN -0.313 -0.849 0.334
Analysis of variance
Analysis of Variance Table of type III with Satterthwaite
approximation for degrees of freedom
Sum Sq Mean Sq NumDF DenDF F.value Pr(>F)
cenAge 22168.6173 22168.6173 1 45.204218 4.4713851 0.040014
CigaretteCurr 14260.3010 14260.3010 1 134.606676 2.8762866 0.092205
cenAge:CigaretteCurr 165.1649 165.1649 1 57.917926 0.0333136 0.855812
Linear mixed model fit by REML
t-tests use Satterthwaite approximations to degrees of freedom ['lmerMod']
Formula: reConc ~ cenAge + CigaretteCurr + (0 + cenAge | HNDid) + (1 | Seq)
Data: exConc
Control: lmerControl(check.nobs.vs.nRE = "warning")
REML criterion at convergence: 1612.9
Scaled residuals:
Min 1Q Median 3Q Max
-1.9238677 -0.4819876 -0.1452019 0.2805955 4.6475664
Random effects:
Groups Name Variance Std.Dev.
HNDid cenAge 1733.367 41.63372
Seq (Intercept) 1435.631 37.88972
Residual 4928.594 70.20394
Number of obs: 139, groups: HNDid, 73; Seq, 31
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 117.26180 14.21455 70.17103 8.24942 0.000000000006239
cenAge -19.20389 8.37700 40.27189 -2.29245 0.027180
CigaretteCurrNo -30.43064 16.13559 131.90789 -1.88593 0.061503
Correlation of Fixed Effects:
(Intr) cenAge
cenAge 0.201
CigarttCrrN -0.682 -0.108
Analysis of variance
Analysis of Variance Table of type III with Satterthwaite
approximation for degrees of freedom
Sum Sq Mean Sq NumDF DenDF F.value Pr(>F)
cenAge 25901.473 25901.473 1 40.272135 5.2553476 0.027180
CigaretteCurr 17529.734 17529.734 1 131.930908 3.5567416 0.061503
Linear mixed model fit by REML
t-tests use Satterthwaite approximations to degrees of freedom ['lmerMod']
Formula: reConc ~ cenAge * dxDiabetes + (0 + cenAge | HNDid) + (1 | Seq)
Data: exConc
Control: lmerControl(check.nobs.vs.nRE = "warning")
REML criterion at convergence: 1678.1
Scaled residuals:
Min 1Q Median 3Q Max
-1.8192568 -0.4740600 -0.1233043 0.3332761 3.7865585
Random effects:
Groups Name Variance Std.Dev.
HNDid cenAge 2596.311 50.95401
Seq (Intercept) 2411.899 49.11109
Residual 3599.304 59.99420
Number of obs: 148, groups: HNDid, 74; Seq, 31
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 90.576608 12.248634 34.445080 7.39483 0.000000013124
cenAge -15.023841 9.669285 43.819900 -1.55377 0.1274328
dxDiabetespreDiabetes 38.447685 18.005770 88.903880 2.13530 0.0354868
dxDiabetesDiabetes 65.796741 31.347244 128.355210 2.09896 0.0377792
cenAge:dxDiabetespreDiabetes -63.765410 21.079856 105.359620 -3.02495 0.0031245
cenAge:dxDiabetesDiabetes -56.519735 29.627355 101.349160 -1.90769 0.0592623
Correlation of Fixed Effects:
(Intr) cenAge dxDbtspD dxDbtsDb cnAg:dxDbtspD
cenAge 0.223
dxDbtsprDbt -0.302 -0.182
dxDibtsDbts -0.220 -0.076 0.155
cnAg:dxDbtspD -0.084 -0.310 -0.103 0.068
cnAg:dxDbtsDb -0.015 -0.257 0.083 -0.366 0.201
Analysis of variance
Analysis of Variance Table of type III with Satterthwaite
approximation for degrees of freedom
Sum Sq Mean Sq NumDF DenDF F.value Pr(>F)
cenAge 61785.931 61785.931 1 67.22876 17.1660782 0.000097886
dxDiabetes 27936.903 13968.451 2 109.43818 3.8808759 0.0235311
cenAge:dxDiabetes 39278.401 19639.200 2 114.65262 5.4563886 0.0054502
Linear mixed model fit by REML
t-tests use Satterthwaite approximations to degrees of freedom ['lmerMod']
Formula: reConc ~ cenAge + dxDiabetes + (0 + cenAge | HNDid) + (1 | Seq)
Data: exConc
Control: lmerControl(check.nobs.vs.nRE = "warning")
REML criterion at convergence: 1704.3
Scaled residuals:
Min 1Q Median 3Q Max
-1.9845595 -0.5177364 -0.1085955 0.2613874 4.5498785
Random effects:
Groups Name Variance Std.Dev.
HNDid cenAge 1918.972 43.80608
Seq (Intercept) 2088.430 45.69934
Residual 4368.684 66.09602
Number of obs: 148, groups: HNDid, 74; Seq, 31
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 86.490687 12.019816 36.997800 7.19567 0.000000015581
cenAge -27.426299 8.496561 42.911200 -3.22793 0.0023913
dxDiabetespreDiabetes 35.501892 18.879866 111.744000 1.88041 0.0626559
dxDiabetesDiabetes 63.024025 29.756530 143.648600 2.11799 0.0358978
Correlation of Fixed Effects:
(Intr) cenAge dxDbtspD
cenAge 0.249
dxDbtsprDbt -0.337 -0.233
dxDibtsDbts -0.248 -0.167 0.211
Analysis of variance
Analysis of Variance Table of type III with Satterthwaite
approximation for degrees of freedom
Sum Sq Mean Sq NumDF DenDF F.value Pr(>F)
cenAge 45519.632 45519.632 1 42.910811 10.4195288 0.0023913
dxDiabetes 29002.771 14501.386 2 127.244987 3.3193943 0.0393326
