There are non-missing telomere length data for 363 participants separated by an average of 4.8 years.
The design is race (African Americans v. whites) × poverty status (below v. above 125% household poverty level) × sex:
, , PovStat = Above
Race
Sex White AfrAm
Women 46 45
Men 44 46
, , PovStat = Below
Race
Sex White AfrAm
Women 44 46
Men 46 46
For preliminary analyses, we're interested in whether change in telomere length is associated with race, poverty status, sex, age, or their interactions.
One way to look at these data is to examine the follow-up time differences (difAge) and change in telomere length (difLen) for each group:
Follow-up duration by sex, race, and poverty status
Sex Race PovStat difAge.n difAge.mean difAge.sd difAge.min difAge.max
1 Women White Above 46 4.2 1.34 0.52 9.0
2 Women White Below 44 4.7 0.77 2.38 6.8
3 Women AfrAm Above 45 4.7 0.90 2.81 8.8
4 Women AfrAm Below 46 5.0 1.22 2.73 9.1
5 Men White Above 44 4.5 1.37 2.24 9.2
6 Men White Below 46 4.9 0.64 3.24 6.5
7 Men AfrAm Above 46 4.5 0.76 2.12 7.1
8 Men AfrAm Below 46 5.9 1.69 3.81 8.9
Change in telomere length by sex, race, and poverty status
Sex Race PovStat difLen.n difLen.mean difLen.sd difLen.min difLen.max
1 Women White Above 46 0.1350 0.63 -1.5 1.4
2 Women White Below 44 -0.0425 0.73 -1.9 1.1
3 Women AfrAm Above 45 -0.0271 0.64 -1.3 1.4
4 Women AfrAm Below 46 -0.0961 0.82 -2.3 1.2
5 Men White Above 44 -0.0020 0.64 -1.2 1.5
6 Men White Below 46 -0.0026 0.85 -2.0 1.7
7 Men AfrAm Above 46 -0.0278 0.61 -1.4 1.3
8 Men AfrAm Below 46 0.0309 0.63 -1.9 1.3
There are no age differences in the sample associated with sex, race, or poverty status:
Association of age at wave 1 with sex, race, and poverty status
Estimate Std. Error t value Pr(>|t|)
(Intercept) 47.952 0.929 51.600 3.89e-168
SexMen 0.421 0.929 0.453 6.50e-01
RaceAfrAm -0.753 0.929 -0.810 4.19e-01
PovStatBelow 0.927 0.929 0.997 3.19e-01
Association of age at wave 3 with sex, race, and poverty status
Estimate Std. Error t value Pr(>|t|)
(Intercept) 52.045 0.933 55.802 5.41e-179
SexMen 0.722 0.933 0.775 4.39e-01
RaceAfrAm -0.319 0.933 -0.342 7.32e-01
PovStatBelow 1.603 0.933 1.718 8.66e-02
However, African Americans have 0.4 years and those below poverty status have 0.7 years greater follow-up duration:
Association of follow-up duration with sex, race, and poverty status
Estimate Std. Error t value Pr(>|t|)
(Intercept) 4.093 0.121 33.83 1.09e-113
SexMen 0.301 0.121 2.49 1.33e-02
RaceAfrAm 0.433 0.121 3.58 3.90e-04
PovStatBelow 0.676 0.121 5.59 4.57e-08
Another way to look at these data is to examine whether change in telomere length (difLen) is correlated with initial age (Age.1) or follow-up time (difAge). This assumes implicitly that time between assays is identical for all participants, which we know is false based on the preceding breakdowns in follow-up times by sex, race, and poverty status.
Age.1 difAge difLen tLength.1 tLength.2
Age.1 1.00 -0.02 0.06 -0.13* -0.08
difAge -0.02 1.00 0.06 0.00 0.07
difLen 0.06 0.06 1.00 -0.53*** 0.46***
tLength.1 -0.13* 0.00 -0.53*** 1.00 0.51***
tLength.2 -0.08 0.07 0.46*** 0.51*** 1.00
n = 363
Neither Age.1 nor difAge are correlated significantly with difLen but these results do not preclude differences in rates of change in telomere length and particularly interactions among sex, race, poverty status, and age. A longitudinal analysis by mixed-model regression is the appropriate technique for examining these effects. This method explicitly includes age differences among the effects.
We start with a fully articulated model (all the effects and all their interactions), after which we will selectively remove nonsignificant interactions by backward elimination.
Analysis of Variance Table of type III with Satterthwaite
approximation for degrees of freedom
Sum Sq Mean Sq NumDF DenDF F.value Pr(>F)
Age 0.9682358 0.9682358 1 511.88944 4.0141085 0.04564763
Sex 3.2126059 3.2126059 1 350.65124 13.3188105 0.00030271
Race 0.2886383 0.2886383 1 350.65124 1.1966358 0.27474600
PovStat 0.6032869 0.6032869 1 350.65124 2.5011049 0.11466800
Age:Sex 0.1435569 0.1435569 1 511.88944 0.5951578 0.44078752
Age:Race 0.0766274 0.0766274 1 511.88944 0.3176818 0.57325099
Age:PovStat 0.1640055 0.1640055 1 511.88944 0.6799333 0.40999377
Sex:Race 2.8319036 2.8319036 1 350.65124 11.7404960 0.00068415
Sex:PovStat 0.0137679 0.0137679 1 350.65124 0.0570789 0.81131341
Race:PovStat 0.2176218 0.2176218 1 350.65124 0.9022157 0.34284249
Age:Sex:Race 0.4978618 0.4978618 1 511.88944 2.0640337 0.15142138
Age:Sex:PovStat 0.0168377 0.0168377 1 511.88944 0.0698056 0.79172641
Age:Race:PovStat 0.5476817 0.5476817 1 511.88944 2.2705769 0.13246767
Sex:Race:PovStat 0.0105366 0.0105366 1 350.65124 0.0436825 0.83456674
Age:Sex:Race:PovStat 0.0033739 0.0033739 1 511.88944 0.0139875 0.90590115
After backward elimination, age, sex, race, and the interaction between sex and race remain in the model (race is nonsignficant but must remain in the model because it participates in the interaction).
Analysis of Variance Table of type III with Satterthwaite
approximation for degrees of freedom
Sum Sq Mean Sq NumDF DenDF F.value Pr(>F)
Age 1.1374498 1.1374498 1 511.95183 4.7158614 0.03034404
Sex 3.2947034 3.2947034 1 357.98743 13.6598250 0.00025337
Race 0.2083356 0.2083356 1 357.96862 0.8637586 0.35331530
Sex:Race 3.0655396 3.0655396 1 357.87281 12.7097127 0.00041313
It's notoriously difficult to interpret these results by just looking at the coefficients. It's far easier to visualize these assocations by plotting the effects.
These plots show the effects in the model expressed over age and are interpreted as an estimate of the age-associated effects on telemere length generated from a repeated measures analysis.
Overall, telomere length decreases significantly with age, but there are no differences in the rate of change associated with sex or race (or poverty status, which dropped out of the analysis). There are overall differences in telomere length by sex such that men have significantly larger telomere lengths than women, but only for African Americans. There are no sex differences for whites (in the figure the plots for white men and women are superimposed).