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Efficient estimation of smooth distributions from coarsely grouped data.

Rizzi S, Gampe J, Eilers PH - Am. J. Epidemiol. (2015)

Bottom Line: Optimal values of the smoothing parameter are chosen by minimizing Akaike's Information Criterion.We demonstrate the performance of this method in a simulation study and provide several examples that illustrate the approach.Wide, open-ended intervals can be handled properly.

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Ungrouping of the age-specific exposure to risk values and estimating age-specific death rates for diseases of the circulatory system, United States, 2009. A) The original exposures taken from the Human Mortality Database were grouped in 5-year bins plus a wide class for ages ≥85 years and an additional bin with 0 counts between ages 115 and 130 years. This histogram was ungrouped with the penalized composite link model. Histogram, original data (black line with overplotted points) and results from ungrouping (solid gray line). The optimal value of the smoothing parameter was log10(λ) = 3.75. B) Death rates obtained from grouped death counts and original exposure to risk values. Death counts taken from the Centers for Disease Control and Prevention Database were grouped in 5-year bins plus an open-ended class for ages ≥85 years. The optimal value of the smoothing parameter was log10(λ) = 5.75. C) Estimated death rates when both the death counts and the exposure numbers were grouped and then ungrouped by the penalized composite link model. The model for the rates had a composition matrix that contained the ungrouped exposures shown in A. The optimal value of the smoothing parameter was log10(λ) = 5.75. In each panel, the original rates (black line with overplotted points) are compared with smoothly estimated rates (solid gray line).
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KWV020F4: Ungrouping of the age-specific exposure to risk values and estimating age-specific death rates for diseases of the circulatory system, United States, 2009. A) The original exposures taken from the Human Mortality Database were grouped in 5-year bins plus a wide class for ages ≥85 years and an additional bin with 0 counts between ages 115 and 130 years. This histogram was ungrouped with the penalized composite link model. Histogram, original data (black line with overplotted points) and results from ungrouping (solid gray line). The optimal value of the smoothing parameter was log10(λ) = 3.75. B) Death rates obtained from grouped death counts and original exposure to risk values. Death counts taken from the Centers for Disease Control and Prevention Database were grouped in 5-year bins plus an open-ended class for ages ≥85 years. The optimal value of the smoothing parameter was log10(λ) = 5.75. C) Estimated death rates when both the death counts and the exposure numbers were grouped and then ungrouped by the penalized composite link model. The model for the rates had a composition matrix that contained the ungrouped exposures shown in A. The optimal value of the smoothing parameter was log10(λ) = 5.75. In each panel, the original rates (black line with overplotted points) are compared with smoothly estimated rates (solid gray line).

Mentions: To demonstrate the performance of the extended approach, we apply it to diseases of the circulatory system and compare estimated and original age-specific death rates. The exposure data were taken from the Human Mortality Database (21). They were provided by single year of age from 0 to 109 years, with a last age class of ≥110 years. We produced ungrouped estimates of the exposures after grouping them in the same age classes as the event counts. Again, the results are reassuring (Figure 4A). In Figure 4B and 4C, 2 versions of estimated death rates are shown. In Figure 4B, only the event counts were ungrouped, but the exposures were taken by single year of age from the Human Mortality Database. In Figure 4C, both the number of events and the exposures were ungrouped. First, the ungrouped exposures were inserted into the composition matrix C, and the rates were estimated from this second penalized composite link model. The model succeeds in producing accurate results not only when the events are binned in intervals but also when the exposure numbers come in age groups. In both cases, the steep decline in death rates after age 0 years counteracts the idea of smoothness that underlies the penalized composite link model; however, this feature could be captured by a single point component.Figure 4.


Efficient estimation of smooth distributions from coarsely grouped data.

Rizzi S, Gampe J, Eilers PH - Am. J. Epidemiol. (2015)

Ungrouping of the age-specific exposure to risk values and estimating age-specific death rates for diseases of the circulatory system, United States, 2009. A) The original exposures taken from the Human Mortality Database were grouped in 5-year bins plus a wide class for ages ≥85 years and an additional bin with 0 counts between ages 115 and 130 years. This histogram was ungrouped with the penalized composite link model. Histogram, original data (black line with overplotted points) and results from ungrouping (solid gray line). The optimal value of the smoothing parameter was log10(λ) = 3.75. B) Death rates obtained from grouped death counts and original exposure to risk values. Death counts taken from the Centers for Disease Control and Prevention Database were grouped in 5-year bins plus an open-ended class for ages ≥85 years. The optimal value of the smoothing parameter was log10(λ) = 5.75. C) Estimated death rates when both the death counts and the exposure numbers were grouped and then ungrouped by the penalized composite link model. The model for the rates had a composition matrix that contained the ungrouped exposures shown in A. The optimal value of the smoothing parameter was log10(λ) = 5.75. In each panel, the original rates (black line with overplotted points) are compared with smoothly estimated rates (solid gray line).
© Copyright Policy - creative-commons
Related In: Results  -  Collection

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getmorefigures.php?uid=PMC4493979&req=5

KWV020F4: Ungrouping of the age-specific exposure to risk values and estimating age-specific death rates for diseases of the circulatory system, United States, 2009. A) The original exposures taken from the Human Mortality Database were grouped in 5-year bins plus a wide class for ages ≥85 years and an additional bin with 0 counts between ages 115 and 130 years. This histogram was ungrouped with the penalized composite link model. Histogram, original data (black line with overplotted points) and results from ungrouping (solid gray line). The optimal value of the smoothing parameter was log10(λ) = 3.75. B) Death rates obtained from grouped death counts and original exposure to risk values. Death counts taken from the Centers for Disease Control and Prevention Database were grouped in 5-year bins plus an open-ended class for ages ≥85 years. The optimal value of the smoothing parameter was log10(λ) = 5.75. C) Estimated death rates when both the death counts and the exposure numbers were grouped and then ungrouped by the penalized composite link model. The model for the rates had a composition matrix that contained the ungrouped exposures shown in A. The optimal value of the smoothing parameter was log10(λ) = 5.75. In each panel, the original rates (black line with overplotted points) are compared with smoothly estimated rates (solid gray line).
Mentions: To demonstrate the performance of the extended approach, we apply it to diseases of the circulatory system and compare estimated and original age-specific death rates. The exposure data were taken from the Human Mortality Database (21). They were provided by single year of age from 0 to 109 years, with a last age class of ≥110 years. We produced ungrouped estimates of the exposures after grouping them in the same age classes as the event counts. Again, the results are reassuring (Figure 4A). In Figure 4B and 4C, 2 versions of estimated death rates are shown. In Figure 4B, only the event counts were ungrouped, but the exposures were taken by single year of age from the Human Mortality Database. In Figure 4C, both the number of events and the exposures were ungrouped. First, the ungrouped exposures were inserted into the composition matrix C, and the rates were estimated from this second penalized composite link model. The model succeeds in producing accurate results not only when the events are binned in intervals but also when the exposure numbers come in age groups. In both cases, the steep decline in death rates after age 0 years counteracts the idea of smoothness that underlies the penalized composite link model; however, this feature could be captured by a single point component.Figure 4.

Bottom Line: Optimal values of the smoothing parameter are chosen by minimizing Akaike's Information Criterion.We demonstrate the performance of this method in a simulation study and provide several examples that illustrate the approach.Wide, open-ended intervals can be handled properly.

View Article: PubMed Central - PubMed

Show MeSH
Related in: MedlinePlus