Multiphasic Analysis of Embryonic Mortality in Chickens1

Multiphasic Analysis of Embryonic Mortality in Chickens1 E. W. JASSIM,* M. GROSSMAN,*^ W. J. KOOPS,t and R.A.J. LUYKXt *Department of Animal Sciences, University of Illinois, Urbana, Illinois 61801, ^Department of Animal Husbandry, Wageningen Agricultural University, P.O. Box 338, 6700 AH Wageningen, The Netherlands, and tHybro R&D, Euribrid B.V., 5830 AA Boxmeer, The Netherlands ABSTRACT Infertility and embryonic mortality are economically important for the commercial broiler industry because they are components of hatchability. Embryonic mortality in chickens is not uniformly distributed over the course of incubation; two phases of embryonic mortality are characteristic of chicken development. The objective of this paper was to develop a mathematical model to assess infertility and to characterize the distribution over time of embryonic mortality in chickens. A model was constructed based on evidence in the literature for multiple phases of embryonic mortality before and during incubation. A multiphasic model, with two phases, included parameters for the proportion of eggs that were infertile and the proportion

that were fertile but the embryo died before or during incubation. For those eggs that were fertile but the embryo died, the model included the proportion of embryonic mortality during each phase, day of peak mortality, and duration of phase. Data on embryonic mortality for white Cornish chickens were used to illustrate the multiphasic model. Model parameters could be estimated easily and interpreted with clear biological meaning. Estimates of parameters, in general, were reasonably precise and consistent with the literature. With multiphasic analysis, one can assess infertility and characterize the distribution of embryonic mortality in chickens, which can lead to a useful understanding of management and genetic aspects of these components of hatchability.

(Key words: multiphasic analysis, infertility, embryonic mortality, hatchability, chicken) 1996 Poultry Science 75:464-471

INTRODUCTION

late phase, with a peak at about Day 19 of incubation (Payne, 1919). Two phases of embryonic mortality are characteristic of avian development (e.g., Byerly, 1931; Bronkhorst, 1933; Insko and Martin, 1935; Romanoff, 1949; Moseley and Landauer, 1949; Landauer, 1951). Phases of embryonic mortality are associated with major changes in embryo metabolism. The first phase coincides with the period of peak lactic acid production (Romanoff, 1949), and occurs during a change in carbon dioxide elimination (Landauer, 1951) and during the time that the mesonephros, part of the embryonic kidney, first functions (Byerly, 1931). The second phase coincides with the period when the demand for oxygen increases significantly (Rol'nik, 1970). Events associated with the second phase of embryonic mortality may even cause d e a t h for several d a y s after h a t c h i n g (Otrygan'eva, 1963). Multiphasic models have been used to study several biological phenomena, including growth (Grossman and Koops, 1988) and egg production (Koops and Grossman, 1992) in chickens and, most recently, nonreturn rates in dairy bulls (Koops et ah, 1995; Grossman et al., 1995). In general, a multiphasic model assumes that a biological process occurs in phases, usually over the course of time. In this study, the model assumes that embryonic mortality occurs in several phases before and during the period of incubation.

Infertility and embryonic mortality are of great economic importance for the commercial broiler industry because they are components of hatchability. Embryonic mortality in chickens is influenced by nutrition (Beer, 1969), management (Byerly, 1938; Lundy, 1969), and genetic factors (Byerly et ah, 1934; Landauer, 1951; Cole, 1969; Hutt, 1969). Embryonic mortality at early (0 to 11 d) and late (12 to 22 d) periods of development has been shown to be lowly heritable (Brah et ah, 1991), as expected for dichotomous traits with one class at low frequency. Embryonic mortality in chickens is not uniformly distributed over the course of incubation. About 65% of embryonic mortality occurs in two phases: an early phase, with a peak at about Day 4 of incubation, and a

Received for publication August 3, 1995. Accepted for publication November 27, 1995. Supported in part by the Illinois Agricultural Experiment Station, Hatch Project 35-323 and Animal Health and Disease Project 35-908, and by a grant to E. W. Jassim by the University of Illinois College of Agriculture JBT Undergraduate Research Scholars Program. Part of this research appeared as an abstract for the 1995 Annual Meeting of the Poultry Science Association. 2 To whom correspondence should be addressed, at 1207 W. Gregory Dr., Urbana, IL 61801.

464

MULTIPHASIC ANALYSIS OF EMBRYONIC MORTALITY : ;

f

465

during incubation (Group 2). Probability of failure to hatch, therefore, is the probability of infertility or embryonic mortality by time t, P(t), which can be written as:

G„

03

P(t) = X P(tlGi)P(Gi)

O

*-<

g

21

Day of incubation FIGURE 1. For eggs that failed to hatch ( ) with probability f, the course of probability for groups: Gj ( ), eggs that were infertile with probability g; and G2 (- • • -), eggs that were fertile but the embryo died with probability (f - g).

Using a multiphasic analysis to study the components of hatchability, infertility, and embryonic mortality, can lead to a useful understanding of management and genetic aspects. The objective of this paper, therefore, was to develop a mathematical model to assess infertility and to characterize the distribution over time of embryonic mortality in chickens. Data on embryonic mortality provided by Euribrid B.V., the Netherlands, were used to illustrate the model.

MATERIALS AND METHODS Mathematical Model All eggs set and incubated can be divided into those that failed to hatch, with probability f, and those that hatched, with probability (1 - f). Those eggs that failed to hatch can be divided further into two groups (Figure 1): Group 1 (Gi), those that were infertile, with probability P(Gi) = g; and Group 2 (G2), those that were fertile but the embryo died either before incubation (before oviposition or during storage) or during incubation, with probability P(G2) = (f - g). Those eggs in Group 1 have a constant probability (g) of being infertile. Those eggs in Group 2, however, have a probability (f - g) of embryonic mortality that is not constant, but can be distributed over phases. With two phases, for example, there may be an early phase, in which the embryo died before incubation or during the early part of incubation, and a late phase, in which the embryo died during the late part of incubation or soon after hatching (Figure 1). To assess infertility and to characterize the distribution of embryonic mortality, we will consider only those eggs that failed to hatch: those that were infertile (Group 1) and those that were fertile but the embryo died before or

[1]

i=l

where P(t I Gj) is the conditional probability of infertility or embryonic mortality by time t, given Group i (i = 1,2). The conditional probability for each group will be examined separately. Group 1: Given that eggs are infertile (Figure 1), the conditional probability of infertility is unity: P(tlGi) = 1

[2]

Group 2: Given that eggs are fertile but the embryo died before or during incubation (Figure 1), the conditional probability of embryonic mortality by time t can be expressed in general as a sum of n logistic functions:

P(tlG2) = X

t - cA

i=l

[3]

1 + e

where n is the number of phases; e is the base of the natural logarithms; and, for phase i, m; is the proportion of embryonic mortality during that phase, subject to the constraint that E" rrtj = 1, so that m n = 1 - E""1 m*; q is the day of peak mortality; and 4d; is a measure in days of duration of the phase, and includes about 96% of mortality during that phase (Koops and Grossman, 1991). Such a function has the property of constraining probabilities within each phase to a lower asymptote of zero for t < q and an upper asymptote of m; for t > q. Because of the possibility that the embryo died before incubation (t < 0), P(tlG2) may not equal zero at t = 0 (Figure 1). Finally, probability of infertility or embryonic mortality by time t, P(t), is modeled by substituting [2] and [3] into [1] to yield the multiphasic model: "i

P(t) = g + (f - g)

i=l

A - Qd 1 + e v i

[4]

Data White Cornish parents used to produce eggs for this study were hatched on 23 March 1994. Between 30 and 33 wk of age, each of four males was allowed to mate in a pen with seven females. For each of two hatches, eggs were collected and stored at 16 C and 75% relative humidity. For Hatch 1, eggs were collected and stored between 18

466

JASSIM ET AL.

October through 31 October; incubation began on 1 November 1994. For Hatch 2, eggs were collected and stored between 1 November through 14 November; incubation began on 15 November 1994. Temperature of the incubator was 37.6 C and relative humidity was 56%. On Day 18 of incubation, eggs were transferred to the hatcher, where temperature was 37.4 C and relative humidity was 57%. On Day 19, relative humidity was increased to 80%. A total of 428 eggs was set, and eggs were candled individually on Days 6, 9, 18, and 21 of incubation. The hatchery manager determined infertility or day of death by breakout analysis of those eggs that were suspected of being infertile or of having suffered embryonic death. Number of infertile eggs and number of eggs containing a dead embryo were recorded by dam, sire, and hatch, for each day of incubation. An egg that was determined to be infertile or that was fertile but the embryo died before incubation was recorded as infertile on Day 0 (Table 1). For an egg that is candled at 7 d or earlier, it is difficult to distinguish with certainty between one that is infertile and one that is fertile but the embryo died before incubation (Munro and Kosin, 1945; Hutt, 1969; North and Bell, 1990). For Day 0 and for each day of incubation, from Day 1 through Day 21, number of eggs that were infertile or that contained a dead embryo was cumulated to obtain the cumulative number (Table 1). Each cumulative number was then divided by total number of eggs set to obtain the relative cumulative number, or cumulative proportion (Table 1). Seven eggs recorded as "pipped" survived through incubation but did not hatch; therefore, they were not included in the analysis. Such eggs were counted as if they hatched, thus causing the estimate of failure to hatch (f) to be biased downward. Infertile eggs that may have developed parthogenetically and died during incubation, however, were included in the analysis. Such eggs, although low in frequency, would be counted as if they were fertile, thus causing the estimate of infertility (g) to be biased downward.

Estimation

of

Parameters

Results from the literature indicate that embryonic mortality in chickens is distributed over two phases. For [4], therefore, we assumed a diphasic (n = 2) model P(t) = 1 - m,

m, t - c,

g + (f - g) 1 + e

ft - c.

1 + e V d2 [5]

where parameters were defined for [4], and where mj is the proportion of embryonic mortality during Phase 1 and

0.25 *» ,0

0.20 0.15

•8

I

o

0.10 0.05

/ ^"B'-e o

0.0 0

3

\

a'^"g*

12

15

18

21

Day of incubation FIGURE 2. Cumulative proportion for failure to hatch for overall (•) and predicted course of probability for failure to hatch for overall ( ). Actual proportion for failure to hatch for overall (O) and predicted probability for failure to hatch for Phase 1 overall ( ) and for Phase 2 overall (- • • -), at each day of incubation.

(1 - mi) is the proportion of embryonic mortality during Phase 2. To assess infertility and to characterize the distribution of embryonic mortality over time, Equation [5] was fitted to cumulative proportion (Table 1), first overall, for a general interpretation, then for each of the two hatches, for a view of the difference between hatches, and finally for each of the four sires, for a view of differences among sires. Parameters were estimated by nonlinear regression, using an adaptive nonlinear least squares algorithm (Sherrod, 1992). A default value of 1 x 10-10 w a s used for the tolerance factor, which specifies the convergence criterion for the iterative estimation procedure. Goodness of fit for the model was measured by residual standard error.

RESULTS AND DISCUSSION Estimates and standard errors for proportion of eggs that failed to hatch (f), proportion of eggs that were infertile (g), proportion of embryonic mortality in Phase 1 (mi), day of peak mortality for Phase 1 (ci), days of duration of Phase 1 (4di), day of peak mortality for Phase 2 fo), and days of duration of Phase 2 (4d2), and the residual standard error are in Table 2, overall, by hatch, and by sire. Each estimate was statistically significant (P < 0.001), except as noted in the table. Cumulative proportion and predicted course for probability of failure to hatch are in Figure 2 for overall, Figure 3 for hatches, and Figures 4 and 5 for sires. To understand and interpret the results better, the distribution for observed proportion and for predicted probability of failure to hatch at each day of incubation are shown in Figure 2. These distributions have two phases, which have peaks of mortality that correspond to the two peaks described in the literature. The accumulation of daily observed proportions results in cumulative proportions, and the accumulation of daily predicted

2 0 2 21 25 25 0.116

5 3 2 20 30 30 0.141

55 55 0.128

55 36 48 76 215

55 43 61 54 213

428

1 2 3 4 Total CN CP

1 2 3 4 Total CN CP

Total CN CP

Sire

Inf. eggs

Eggs set

2

3

4

5

6

7

0 0 2 0

1 0 1 2

0 0 0 0

0 1 1 0

0 0 0 0

0 0 0 0

6 5 11 2 2 3 0 61 66 77 79 81 84 84 0.142 0.154 0.180 0.185 0.189 0.196 0.196

32 34 38 38 40 40 40 0.150 0.160 0.178 0.178 0.188 0.188 0.188

0 0 0 2

1 0 3 0 0 0 0 1 0 1 0 2 0 0 2 1 1 0 0 0 0 2 1 1 0 0 3 0 4 3 7 2 0 3 0 29 32 39 41 41 44 44 0.135 0.149 0.181 0.191 0.191 0.205 0.205

1

9

10

11

12

(Hatch 1)

13

14

0 0 0 0

0 0 0 0

- HatclK 2 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0

0 0 0 1 0 0 0 84 84 84 85 85 85 85 0.196 0.196 0.196 0.199 0.199 0.199 0.199

Overall

40 40 40 40 40 40 40 0.188 0.188 0.188 0.188 0.188 0.188 0.188

0 0 0 0

0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 44 44 44 45 45 45 45 0.205 0.205 0.205 0.209 0.209 0.209 0.209

8

Day of incubation

0 0 0

16

1 0 86 86 0.201 0.201

0 0 0 0 0 0 0 0 0 0 40 40 0.188 0.188

o 1 o 0 46 46 0.214 0.214

0 1 0

15

TABLE 1. Number of eggs set, number of infertile eggs (Inf. eggs), number of dead embryos for each day of by sire, by hatch, and overall; and cumulative number (CN) and cumulative proportion (CP), by hatch and

JASSIM ET AL.

468

0.25

0.45 r ><)•*

%

/•••

_p.-G-©-c-cr Q..o..a-e>

0.20 \

f.

o

£ x>

•>

a

t-i

0.15

«.*••*

0.40

0.35 f

>

v

/

0.10

0.30 12

15

18

21

12

Day of incubation

15

18

21

Day of incubation

FIGURE 3. Cumulative proportion for failure to hatch for Hatch 1 (O) and Hatch 2 (•), and predicted course of probability for failure to hatch for Hatch 1 ( ) and Hatch 2 ( ).

FIGURE 5. Cumulative proportion for failure to hatch for Sire 4 (•) and predicted course of probability for failure to hatch for Sire 4

probability results in the predicted course of probability for failure to hatch.

observed proportion includes those eggs that were infertile as well as those that were fertile but the embryo died before incubation. Thus, the difference (0.013) between the observed and the estimated proportion of infertile eggs is a measure of the proportion of eggs that were fertile but the embryo died before incubation. Proportion of eggs that were fertile but the embryo died before or during incubation (f - g) was 0.11. For those eggs (Figure 2), about 0.70 (mi) of the embryos died during Phase 1, the early phase, which means that about 0.30 (1 mi) died during Phase 2, the late phase. Peak mortality was at 1.9 d (ci) for the early phase and at 18.4 d fo) for the late phase. Duration was about the same for each phase: 4.6 d (4di) for the early phase and about 4.8 d (4d2) for the late phase.

Overall Overall (Table 2 and Figure 2), proportion of eggs that failed to hatch (f) was about 0.23, which means that about 77% of eggs hatched. Proportion of eggs that were infertile (g) was about 0.12, which means that about 88% of eggs were fertile. An estimate for proportion of eggs that were infertile (0.115, Table 2) is a measure of the "true" proportion of infertile eggs and not a measure of the observed proportion of "infertile" eggs (0.128, Table 1), because the

(---)•

Hatches U.10

sZ •

J—j

0.12 / /'•

n

4..^-*-*—*-*-*—jr-C'C

/ / / O S-l

OH

0.08

.ojf

<

0.04 , "/•

12

15

18

21

Day of incubation FIGURE 4. Cumulative proportion for failure to hatch for Sire 1 (o), Sire 2 (A), and Sire 3 (•); predicted course of probability for failure to hatch for Sire 1 ( ), Sire 2 ( ), and Sire 3 ( ).

Proportion of eggs that failed to hatch (f) was about 0.25 for Hatch 1 and about 0.22 for Hatch 2 (Table 2 and Figure 3). Proportion of eggs that were infertile (g) was about 0.10 for Hatch 1 and about 0.14 for Hatch 2. The difference between the observed (Table 1) and the estimated (Table 2) proportion of infertile eggs was 0.02 for Hatch 1 and 0.005 for Hatch 2. Although the difference between hatches is fourfold, it is of little practical importance. Proportion of eggs that were fertile but failed to hatch (f - g) was about 0.15 for Hatch 1 and about half of that (0.08) for Hatch 2. For those eggs (Figure 3), the proportion of embryos that died during the early phase of incubation (mi) was about 0.73 for Hatch 1 but only about 0.63 for Hatch 2. Consequently, the proportion of embryos that died during the late phase (1 - mi) was about 0.27 for Hatch 1 and about 0.37 for Hatch 2. Thus, most embryonic mortality occurred during the early phase compared with the late phase, but more so for Hatch 1 than for Hatch 2.

469

MULTIPHASIC ANALYSIS OF EMBRYONIC MORTALITY 1

TABLE 2. Estimates (Est) and standard errors (SE) for proportion of eggs that failed to hatch (f), proportion of eggs that were infertile (g), proportion of embryonic mortality in phase 1 taj), day of peak mortality for phase 1 (q), days of duration of phase 1 (4d t ), day of peak mortality for phase 2 (c2), and days of duration of phase 2 (4d2), and the residual standard error, overall, by hatch, and by sire Hatch Parameter f

g mj

Cl

4di c

2

4d 2 Residual

Est SE Est SE Est SE Est SE Est SE Est SE Est SE SE

Sire

Overall

1

2

1

2

3

0.233 0.005 0.115 0.008 0.696 0.044 1.90 0.29 4.60 0.66 18.44 0.51 4.82" 1.35 0.0024

0.247 0.009 0.096 0.014 0.730 0.063 1.84 0.40 4.97 0.90 17.78 0.95 6.67* 2.60 0.0036

0.218 0.001 0.136 0.005 0.628 0.035 2.00 0.24 3.67 0.64 18.88 0.24 2.73" 0.75 0.0021

0.164 0.010 0.068 0.003 0.449 0.059 2.41 0.24 0.77+ 0.41 17.79 0.85 6.66" 2.13 0.0043

0.163 0.010 0.040 0.004 0.498 0.050 2.88 0.18 2.86 0.61 18.40 0.61 5.75" 1.46 0.0036

0.130 0.005 0.029 0.005 0.798 0.043 2.05 0.14 2.76 0.43 18.66 0.66 3.37"s 1.96 0.0031

4 0.432 0.037 0.031^ 0.767 0.924 0.186 -S.OO"5 7.47 9.82* 3.47 20.14 2.68 3.90"5 4.62 0.0047

'Each estimate is statistically significant at the 0.001 level of probability, except as noted. + P < 0.10. •P < 0.05. " P < 0.01.

Peak mortality during the early phase (ci) was at about 1.8 d for Hatch 1 and at 2.0 d for Hatch 2. Peak mortality during the late phase fo) was at about 17.8 d for Hatch 1 and at about 18.9 d for Hatch 2. Duration of the early phase (4dj) was about 5 d for Hatch 1 and about 3.7 d for Hatch 2. Duration of the late phase (4d2) was about 6.7 d for Hatch 1, but about 2.7 d for Hatch 2. Note that the standard error for duration of the late phase for Hatch 1 is larger than those for other durations. This relatively large standard error may be due to a smaller proportion of embryonic deaths in the late phase of Hatch 1 distributed over a longer period of time (Figure 3). The model fitted Hatch 2 better (residual SE = 0.0021) than Hatch 1 (residual SE = 0.0036).

Sires For infertility and for many characteristics of embryonic mortality, results for Sire 4 deviated from results for the other three sires (Tables 1 and 2); thus, results for Sire 4 will be discussed in comparison with average results for the other sires. Results for Sires 1,2, and 3 are in Figure 4, whereas results for Sire 4 are in Figure 5. Proportion of eggs that failed to hatch (f) averaged about 0.15 for Sires 1, 2, and 3 (Table 2 and Figure 4) and was about 0.43 for Sire 4 (Table 2 and Figure 5). Proportion of eggs that were infertile (g) averaged about 0.05 and was about 0.03 for Sire 4, which was not significantly different from zero. The difference between the observed (computed from Table 1) and the estimated (Table 2)

proportion of infertile eggs averaged 0.001 and was 0.284 for Sire 4. There were almost no eggs that were fertile but the embryo died before incubation for Sires 1, 2, and 3, whereas this proportion was almost 30% for Sire 4. For the proposed model, the high estimate for Sire 4 may be due to the high number of dead embryos observed on Day 1 relative to the number observed on Day 2 (Table 1 and Figure 5), in contrast to the numbers observed for Sires 1,2, and 3 (Table 1 and Figure 4). This means that the slope of the curve at Day 1 is steeper for Sire 4 than for the other sires, which results in a higher estimate of the proportion of dead embryos before incubation for Sire 4 than for the other sires. The higher number of dead embryos observed on Day 1 for Sire 4 may be due, in part, to the difficulty in determining whether an embryo died before incubation, on Day 1, or on Day 2. Proportion of eggs that were fertile but failed to hatch (f - g) averaged about 0.1 for Sires 1,2, and 3 (Table 2 and Figure 4), but was about 0.4 for Sire 4 (Table 2 and Figure 5). For those eggs, the proportion of embryos that died during the early phase of incubation (mi) averaged about 0.58, but was about 0.92 for Sire 4. Consequently, the proportion that died during the late phase (1 - mi) averaged about 0.42, but was only about 0.08 for Sire 4. Again, most embryonic mortality occurred during the early phase compared with the late phase, but more so for Sire 4 than for the other sires. Peak mortality during the early phase (ci) was on average at about 2.5 d for Sires 1, 2, and 3 (Table 2 and Figure 4), but at -3.0 d with a high standard error for Sire 4 (Table 2 and Figure 5). The reason for this negative

470

JASSIM ET Ah.

estimate is that the steep slope of the curve at Day 1 for Sire 4, as mentioned above, resulted in an earlier (before incubation) estimate of day at peak mortality for Sire 4 than for the other sires. Peak mortality during the late phase (C2) was on average at about 18.3 d and at about 20.1 d for Sire 4. Duration of the early phase (4di) averaged about 2.1 d for Sires 1, 2, and 3 (Table 2 and Figure 4) and about 9.8 d for Sire 4 (Table 2 and Figure 5). Duration of the late phase (4d2) averaged about 5.3 d, but about 3.9 d for Sire 4. Estimates of duration for Sire 4 were associated with larger standard errors than estimates of duration for the other sires. Larger standard errors for Sire 4 may be due to a smaller proportion of embryonic deaths in the late phase distributed over a longer period of time (Figure 5). The model fitted Sires 1, 2, and 3 better (average residual SE = 0.0037) than Sire 4 (residual SE = 0.0047). Note that residual standard errors tended to increase as number of observations per classification decreased. In this study, estimates of parameters overall, for hatches, and for sires were reasonably precise and consistent, except for Sire 4. Low numbers of observations for embryonic mortality increases the risk that the observed distribution of embryonic mortality, between and within phases, will deviate from the expected distribution under the proposed model, as seen for Sire 4. For hatch-sire classifications, for example, it is impossible to fit the late phase for Sires 3 and 4 in Hatch 1, because there are no observations (Table 1). Sire 4 adversely affected the results overall and for hatches, obscuring the "true" assessment of infertility and characterization of the distribution of embryonic mortality. Consequently, results overall and for hatches, which include results for Sire 4, are not in good agreement with the literature. If data for Sire 4 were excluded, however, results overall, for hatches, and for sires are consistent with the literature. Despite the small data set used in this study, the proposed model enables one to assess infertility and to characterize embryonic mortality for hatches or for sires.

Conclusions The objective of this paper was to develop a mathematical model to assess infertility and to characterize the distribution over time of embryonic mortality in chickens. A model was constructed based on evidence in the literature for multiple phases of embryonic mortality before and during incubation. A multiphasic model, with two phases, included parameters for the proportion of eggs that failed to hatch and the proportion that were infertile. For those eggs that were fertile but the embryo died before or during incubation, the model included the proportion of embryonic mortality during each phase, day of peak mortality, and duration of phase. Data on embryonic mortality for white Cornish chickens were used to illustrate the multiphasic model. Model parameters could be estimated easily and inter-

preted with clear biological meaning. Estimates of parameters, in general, were reasonably precise and consistent. Low numbers of observations for embryonic mortality increases the risk that observed distribution of embryonic mortality, between and within phases, will deviate from the expected distribution under the proposed model. For example, it is impossible to fit the late phase if embryonic mortality occurs only during the early phase. With multiphasic analysis, one can assess infertility and characterize the distribution of embryonic mortality in chickens, which can lead to a useful understanding of the management and genetic aspects of these components of hatchability.

ACKNOWLEDGMENTS Cor Aldenzee, Euribrid B. V., collected the data for this study, and Rohan L. Fernando, University of Illinois, provided helpful discussions. Students in a course on scientific writing taught at Wageningen Agricultural University, the Netherlands, provided useful suggestions to improve the first draft of the manuscript. This paper is dedicated to the memory of B. B. Bohren, mentor of M. Grossman and Professor Emeritus of Genetics, Purdue University Department of Animal Sciences, who dedicated part of his professional career to studying the genetics of hatchability in chickens.

REFERENCES Beer, A. E., 1969. A review of the effects of nutritional deficiencies on hatchability. Pages 93-108 in: The Fertility and Hatchability of the Hen's Egg. T. C. Carter and B. M. Freeman, ed. Oliver and Boyd, Edinburgh, UK. Brah, G. S., J. S. Sandhu, and M. L. Chaudhary, 1991. Heritability of components of incubation mortality in White Leghorns. Br. Poult. Sci. 32:871-874. Bronkhorst, J. J., 1933. Physical, chemical, and physiological study of high and low hatching lines of single comb White Leghorns. Ph.D. Thesis, Cornell University. Page 315 in: The hatchability of chicken eggs as influenced by environment and heredity. Bulletin 262, revised. W. Landauer, 1951. Storrs Agricultural Experiment Station, University of Connecticut, Storrs, CT. Byerly, T. C, 1931. Time of occurrence and probable causes of mortality in chick embryos. Pages 178-186 in: Proceedings 4th World's Poultry Cong. London, UK. Byerly, T. C, 1938. Effect of different incubation temperatures on mortality of chick embryos. Poultry Sci. 17:200-205. Byerly, T. C, C. W. Knox, and M. A. Jull, 1934. Some genetic aspects of hatchability. Poultry Sci. 13:230-238. Cole, R. K., 1969. Lethal and semi-lethal genes and their effects on embryonic development and the production of normal chicks. Pages 57-69 in: The Fertility and Hatchability of the Hen's Egg. T. C. Carter and B. M. Freeman, ed. Oliver and Boyd, Edinburgh, UK. Grossman, M., and W. J. Koops, 1988. Multiphasic analysis of growth curves in chickens. Poultry Sci. 67:33-42. Grossman, M., W. J. Koops, and J.H.G. den Daas, 1995. Multiphasic analysis of reproductive efficiency in dairy bulls. J. Dairy Sci. 78:2871-2876.

MULTIPHASIC ANALYSIS OF EMBRYONIC MORTALITY Hutt, F. B., 1969. Genetic aspects of fertility and hatchability in the fowl. Pages 49-56 in: The Fertility and Hatchability of the Hen's Egg. T. C. Carter and B. M. Freeman, ed. Oliver and Boyd, Edinburgh, UK. Koops, W. J., and M. Grossman, 1991. Applications of a multiphasic growth function to body composition in pigs. J. Anim. Sci. 69:3265-3273. Koops, W. J., and M. Grossman, 1992. Characterization of poultry egg production using a multiphasic approach. Poultry Sci. 71:399-405. Koops, W. J., M. Grossman, and J.H.G. den Daas, 1995. A model for reproductive efficiency in dairy bulls. J. Dairy Sci. 78:921-928. Insko, W. M., Jr., and J. H. Martin, 1935. Mortality of the turkey embryo. Poultry Sci. 14:361-364. Landauer, W., 1951. The Hatchability of Chicken Eggs as Influenced by Environment and Heredity. Bulletin 262, revised. Storrs Agricultural Experiment Station, University of Connecticut, Storrs, CT. Lundy, H., 1969. A review of the effects of temperature, humidity, turning and gaseous environment in the incubator on the hatchability of the hen's egg. Pages 143-176 in: The Fertility and Hatchability of the Hen's Egg. T. C. Carter and B. M. Freeman, ed. Oliver and Boyd, Edinburgh, UK.

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