Ecological Archives E096-005-A3

Vérane Berger, Jean-Franҫois Lemaítre, Jean-Michel Gaillard, and Aurélie Cohas. 2015. How do animals optimize the size–number trade-off when aging? Insights from reproductive senescence patterns in marmots. Ecology 96:46–53. http://dx.doi.org/10.1890/14-0774.1

Appendix C. Detailed statistical analyses.

1. Analyses of confounding variables

Offspring mass, litter size, post-weaning survival and annual reproductive success depend on environmental factors varying in time and space. In mountains, environmental conditions vary sharply from one year to the next. Thus, reproductive traits are likely also to vary from one year to the next. Moreover, the location of a marmot's territory (i.e., in the valley, oriented South or North) affects environmental factors. For instance, the North aspect territories have the lowest food resources (Allainé et al. 1998), so that mothers from these territories are expected to be lighter and to produce fewer offspring of lower quality than females in the valley or in the South. The age of offspring influences offspring mass (Allainé et al. 1998) and can prevent the detection of age-dependent variation in offspring mass at weaning if not accounted for. Moreover, the number of male subordinates influences offspring survival during hibernation as well as annual reproductive success. As all observed take-overs by males were followed by the infanticide of the entire litter, the displacement of the dominant male must also be included in the analyses of offspring survival and annual reproductive success. We took into account all these potentially confounding effects, retained the statistically significant ones, and used the residual variation to assess the litter-size body-size trade-off.

Table C1. Baseline models for offspring mass (a), litter size (b), post-weaning survival (c) and annual reproductive success (d). We provide the equation of the full model and the baseline models obtained after selection of the confounding variables. F(variable) indicates that the variable is entered as a factor in the model. (1|variable) indicates that the variable is entered as a random effect on the intercept. We used Wald's test to measure the contribution of each categorical factor to the overall model, and include the statistic (χ²), the degrees of freedom (df) and the p value (P).

(a) Offspring mass

Full model

Offspring mass = F(year) + F(aspect of the territory) + age of offspring + (1|mother) + (1|group)+ (1|id)

Baseline model

Offspring mass = F(year) + age of offspring + (1|mother) + (1|group)+ (1|id)

 

Parameters of the baseline model

Dependent variable

Distribution

Link function

Explanatory variable

β

SE

P

Wald test

χ²

df

P

Offspring mass

Normal

Identity

Intercept

417.20

85.38

<0.01

24.80

1

<0.01

Year      

1994

-13.05

102.09

0.90

19.50

18

0.36

1995

-97.41

88.48

0.27

1996

-54.83

93.03

0.56

1997

-8.52

89.32

0.92

1998

12.47

82.53

0.88

1999

-17.47

82.11

0.83

2000

-64.16

84.03

0.44

2001

-10.52

78.93

0.89

2002

-64.98

81.18

0.42

2003

25.60

79.48

0.75

2004

-3.95

78.89

0.96

2005

 30.26

77.36

0.70

2006

-43.78

83.45

0.60

2007

40.24

78.43

0.61

2008

45.56

79.48

0.57

2009

12.78

78.34

0.87

2010

-21.86

77.62

0.78

2011

25.50

77.73

0.74

Age of offspring

15.04

0.94

<0.01

267.10

1

<0.01

 

(b) Litter size

Full model

Litter size = F(year) + F(aspect of the territory ) + (1|mother)

Baseline model

Litter size = F(year) + F(aspect of the territory ) + (1|mother)

 

Parameters of the baseline model

Dependent variable

Distribution

Link function

Explanatory variable

β

SE

P

Wald test

χ²

df

P

Litter size

Poisson

Log

Intercept

1.46

0.18

<0.01

69.5

1

<0.01

Year

1994

-0.12

0.24

0.60

17.00

18

0.53

1995

-0.12

0.25

0.62

1996

-0.23

0.25

0.37

1997

-0.05

0.22

0.82

1998

-0.22

0.21

0.29

1999

-0.27

0.22

0.20

2000

-0.08

0.22

0.73

2001

-0.30

0.21

0.16

2002

-0.39

0.22

0.08

2003

-0.27

0.21

0.20

2004

-0.28

0.21

0.20

2005

-0.32

0.21

0.12

2006

-0.34

0.22

0.13

2007

-0.20

0.21

0.34

2008

-0.34

0.21

0.10

2009

0.16

0.20

0.43

2010

-0.34

0.21

0.10

2011

-0.32

0.21

0.13

Aspect of the territory

South 

0.14

0.05

0.01

7.4

1

<0.01

 

(c) Post-weaning survival

Full model

Survival = F(year) + F(aspect of the territory) + change of dominant male + number of helpers + (1|mother)

Baseline model

Survival = F(year) + F(aspect of the territory) + change of dominant male + number of helpers + (1|mother)

 

Parameters of the baseline model

Dependent variable

Distribution

Link function

Explanatory variable

β

SE

P

Wald test

χ²

df

P

Post-weaning survival

Binomial

Logit

Intercept

1.81

1.04

0.08

3.4

1

0.07

Year

1994

-0.19

1.24

0.88

33.5

17

<0.01

1995

-0.67

1.24

0.59

1996

-1.10

1.32

0.41

1997

-1.58

1.13

0.16

1998

-0.64

1.12

0.57

1999

-0.19

1.17

0.87

2000

-1.61

1.16

0.17

2001

 -1.18

 1.13

0.30

2002

-1.78

1.16

0.13

2003

-1.05

1.12

0.35

2004

-1.74

1.14

0.13

2005

-0.57

1.12

0.61

2006

-1.05

1.16

0.36

2007

-0.87

1.11

0.43

2008

-0.85

1.11

0.44

2009

-2.19

1.11

0.05

2010

-2.27

1.11

0.04

2011

-0.59

1.11

0.60

Aspect of the territory

South

-0.98

0.26

<0.01

16.1

1

<0.01

Change of male dominant

-0.78

0.27

<0.01

9.0

1

<0.01

Number of helpers

0.22

0.12

0.06

0.32

1

0.06

 

 (d) Annual reproductive success

Full model

Annual reproductive success = F(year) + F(aspect of the territory) + change of dominant male + number of helpers + (1|mother)

Baseline model

Annual reproductive success = F(year) + F(aspect of the territory) + change of dominant male + number of helpers + (1|mother)

 

Parameters of the baseline model

Dependent variable

Distribution

Link function

Explanatory variable

β

SE

P

Wald test

χ²

df

P

Annual reproductive success

Poisson

Log

Intercept

1.31

0.32

<0.01

18.2

1

<0.01

Year

1994

-0.12

0.43

0.78

26.8

18

0.08

1995

-0.24

0.47

0.61

1996

-0.59

0.47

0.21

1997

-0.64

0.43

0.14

1998

-0.37

0.38

0.32

1999

-0.35

0.39

0.37

2000

-0.61

0.43

0.16

2001

-0.66

0.40

0.10

2002

-0.97

0.45

0.03

2003

-0.61

0.39

0.11

2004

-0.80

0.41

0.05

2005

-0.45

0.37

0.23

2006

-0.62

0.42

0.14

2007

-0.42

0.37

0.26

2008

0.55

0.38

0.14

2009

-1.07

0.41

0.01

2010

-1.23

0.42

<0.01

2011

-0.44

0.38

0.25

Aspect of the territory

South

-0.20

0.11

0.08

3.4

1

0.06

Change of male dominant

-0.35

0.15

0.02

6.1

1

0.02

Number of helpers

0.10

0.05

0.05

4.3

1

0.05

 

2. Age-specific models of reproductive traits

Table C2. Age-specific models of offspring mass, litter size, litter mass, offspring mass corrected for litter size, post-weaning survival and annual reproductive success in female Alpine marmots (Marmota marmota) at La Grande Sassière (French Alps) fitted using Generalized Additive Mixed Models. Base corresponds to the baseline model. Age and Age² correspond to linear and quadratic relationship, respectively, between age and the focal reproductive trait. F(Age) corresponds to the full age-dependent model (i.e., age included as a factor). S(age) corresponds to a smoothed term of age. T(5) corresponds to a trait value linearly increasing until 5 years of age and then remaining constant with age. T(10) corresponds to a trait value constant until 10 years of age and then a linear decrease with increasing age. The number of parameters is indicated by k. The best model including an age effect is in bold and the best model overall is highlighted with gray shading.

Trait

Model

k

AIC

ΔAIC

AICw

Offspring mass (N = 549)

Base

25

6058.02

4.19

0.06

Base + Age

26

6055.47

1.65

0.22

Base + Age²

27

6057.26

3.43

0.09

Base + T(10)

27

6058.18

4.35

0.06

Base + F(age)

37

6053.83

0.00

0.49

Base + S(age)

26

6057.47

3.65

0.08

Litter size (N = 202)

Base

22

173.18

3.52

0.10

Base + Age

23

174.25

4.59

0.06

Base + Age²

24

172.10

2.43

0.18

Base + T(10)

24

169.66

0.00

0.62

Base + F(age)

34

179.91

10.25

<0.01

Base + S(age)

23

176.25

6.60

0.02

Litter mass (N = 136)

Base

2

2073.75

3.88

0.08

Base + Age

3

2074.33

4.46

0.06

Base + Age²

4

2071.39

1.52

0.27

Base + T(10)

4

2069.87

0.00

0.57

Base + F(age)

13

2084.56

14.69

<0.01

Base + S(age)

3

2076.41

6.54

0.02

Offspring mass corrected for litter size (N = 549)

Base

29

6006.65

0.26

0.23

Base + Age

30

6006.39

0.00

0.26

Base + Age²

31

6007.73

1.34

0.14

Base + T(5)

31

6006.71

0.31

0.23

Base + F(age)

41

6010.28

3.89

0.04

Base + S(age)

30

6008.39

2.00

0.10

Post-weaning survival (N =  725)

Base

23

789.07

0.00

0.56

Base + Age

24

791.09

2.01

0.20

Base + Age²

25

792.61

3.48

0.01

Base + T(5)

24

793.18

4.10

0.07

Base + F(age)

36

879.48

90.40

<0.01

Base + S(age)

25

793.18

4.02

0.07

Annual reproductive success (N = 202)

Base

22

502.04

2.74

0.16

Base + Age

23

503.93

4.63

0.06

Base + Age²

24

499.23

0.00

0.62

Base + T(10)

24

502.37

3.07

0.13

Base + F(age)

34

546.56

47.26

<0.01

Base + S(age)

23

505.93

6.63

0.02

 

 

Table C3a. Age-specific models of offspring mass, litter size, litter mass, offspring mass corrected for litter size, post-weaning survival and annual reproductive success in female Alpine marmots (Marmota marmota) at La Grande Sassière (French Alps) fitted using Generalized Additive Mixed Models. Longevity and last year effect (LYE) are added to the two best models of age-specific variation. Base corresponds to the baseline model.T(10) corresponds to a constant trait value until 10 years of age followed by a linear decrease with increasing age. The number of parameters is indicated by k. The selected model is highlighted in gray. Data sets of longevity and LYE are reduced because we excluded 35 females that were alive when the study ended.

Trait

Model

k

AIC

ΔAIC

AICw

Offspring mass (N = 304)

Base + Age

23

3373.50

0.95

0.21

Base + Age + longevity

24

3372.55

0.00

0.33

Base + Age + LYE

24

3372.91

0.35

0.28

Base + Age + longevity + LYE

25

3373.83

1.28

0.18

Litter size (N = 109)

Base + T(10)

24

113.08

0.00

0.56

Base + T(10) + longevity

25

155.30

2.23

0.18

Base + T(10) + LYE

25

115.33

2.25

0.18

Base + T(10) + longevity + LYE

26

117.22

4.15

0.07

Litter mass (N = 75)

Base + T(10)

4

1149.43

0.00

0.44

Base + T(10) + longevity

5

1151.73

1.98

0.16

Base + T(10) + LYE

5

1151.43

2.00

0.16

Base + T(10) + longevity + LYE

6

1150.73

1.30

0.23

Offspring mass corrected for litter size (N = 304)

Base

29

3347.73

0.00

0.49

Base + longevity

30

3349.59

1.85

0.20

Base + LYE

30

3373.81

1.94

0.18

Base + longevity + LYE

31

3350.26

2.54

0.14

Post-weaning survival (N = 384)

 

Base

23

515.57

0.00

0.58

Base + longevity

24

517.79

2.22

0.19

Base + LYE

24

517.88

2.31

0.18

Base + longevity + LYE

25

520.99

5.42

0.04

Annual reproductive success (N = 109)

Base + Age²

23

307.23

0.00

0.47

Base + Age²+ longevity

24

308.89

1.66

0.20

Base + Age² + LYE

24

308.73

1.50

0.21

Base + Age²+ longevity + LYE

25

310.08

2.85

0.11

 

Table C3b. Summary of age-specific models including not statistically significant effects of longevity and last year effect (LYE) on offspring mass, litter size, litter mass, offspring mass corrected for litter size, post-weaning survival and annual reproductive success in female Alpine marmots (Marmota marmota) at La Grande Sassière (French Alps). Est. corresponds to the parameter estimate. SE corresponds to the standard error of the parameter estimate. P corresponds to the p value indicating the statistical significance of the effect of longevity or last year effect (LYE). See Table 3a for details about the model selected for each trait.

TableC3b

Literature cited

Allainé, D., L. Graziani, and J. Coulon. 1998. Postweaning mass gain in juvenile Alpine marmots Marmota marmota.Oecologia 113:370–376.

[Back to E096-005]