If we condense the data completely (ie average over the results on each day) we have a data frame that looks like this
> cdatadaynum
biorep daynight treatment count nflies
1 1 day cs 12620 2662
2 1 day trh 8371 1509
3 1 night cs 10820 1972
4 1 night trh 6132 987
5 2 day cs 18783 4520
6 2 day trh 25412 4820
7 2 night cs 12897 2237
8 2 night trh 11446 1834
9 3 day cs 18449 4247
10 3 day trh 21226 3329
11 3 night cs 9597 1740
12 3 night trh 8356 1398
13 4 day cs 17858 4549
14 4 day trh 13668 2926
15 4 night cs 13580 2525
16 4 night trh 9861 1605
17 5 day cs 14132 3498
18 5 day trh 20023 4165
19 5 night cs 8392 1604
20 5 night trh 18655 3076
> This is easy to analyze using a GLM with nflies as the offset
> gm1=glm(count~offset(log(nflies))+treatment+daynight,cdatadaynum,family=poisson)And the results look like this
> summary(gm1)
Call:
glm(formula = count ~ offset(log(nflies)) + treatment + daynight,
family = poisson, data = cdatadaynum)
Deviance Residuals:
Min 1Q Median 3Q Max
-12.579 -5.726 -1.290 3.974 29.411
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 1.460224 0.003090 472.50 <2e-16 ***
treatmenttrh 0.183433 0.003779 48.54 <2e-16 ***
daynightnight 0.204349 0.003870 52.80 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
(Dispersion parameter for poisson family taken to be 1)
Null deviance: 6879.7 on 19 degrees of freedom
Residual deviance: 1747.9 on 17 degrees of freedom
AIC: 1980.3
Number of Fisher Scoring iterations: 4As it happens we can get an almost identical result via a much fancier statistical route. A Generalized Linear Mixed Effects Model. This performs no averaging across the days for a given experiment, but instead treats days as a random effect nested within biological replicates.
> gmr1=glmer(count~offset(log(nflies))+treatment+daynight+(1|biorep)+(1|biorep:daynum),cdata,family=poisson) # Wins in AIC tests
> summary(gmr1)
Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
Family: poisson ( log )
Formula: count ~ offset(log(nflies)) + treatment + daynight + (1 | biorep) + (1 | biorep:daynum)
Data: cdata
AIC BIC logLik deviance df.resid
2122.7 2133.1 -1056.3 2112.7 55
Scaled residuals:
Min 1Q Median 3Q Max
-7.9337 -3.7030 -0.1025 2.8903 16.3533
Random effects:
Groups Name Variance Std.Dev.
biorep:daynum (Intercept) 0.0004911 0.02216
biorep (Intercept) 0.0028091 0.05300
Number of obs: 60, groups: biorep:daynum, 15; biorep, 5
Fixed effects:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 1.459669 0.024584 59.38 <2e-16 ***
treatmenttrh 0.190293 0.003834 49.64 <2e-16 ***
daynightnight 0.209005 0.003889 53.74 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Correlation of Fixed Effects:
(Intr) trtmnt
treatmnttrh -0.078
daynghtnght -0.062 -0.006Also a model with day-night interaction effect
> gmr1=glmer(count~offset(log(nflies))+treatment*daynight+(1|biorep)+(1|biorep:daynum),cdata,family=poisson) # Wins in AIC tests
> summary(gmr1)
Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
Family: poisson ( log )
Formula: count ~ offset(log(nflies)) + treatment * daynight + (1 | biorep) + (1 | biorep:daynum)
Data: cdata
AIC BIC logLik deviance df.resid
1906.6 1919.2 -947.3 1894.6 54
Scaled residuals:
Min 1Q Median 3Q Max
-8.6709 -3.9065 0.3286 2.3405 14.4777
Random effects:
Groups Name Variance Std.Dev.
biorep:daynum (Intercept) 0.0004911 0.02216
biorep (Intercept) 0.0027018 0.05198
Number of obs: 60, groups: biorep:daynum, 15; biorep, 5
Fixed effects:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 1.436783 0.024197 59.38 <2e-16 ***
treatmenttrh 0.234689 0.004874 48.15 <2e-16 ***
daynightnight 0.267185 0.005520 48.40 <2e-16 ***
treatmenttrh:daynightnight -0.114671 0.007766 -14.77 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Correlation of Fixed Effects:
(Intr) trtmnt dynght
treatmnttrh -0.103
daynghtnght -0.092 0.450
trtmnttrh:d 0.065 -0.617 -0.709