# Does the parametric coefficient from GAMM compare patterns between groups over time, or just the means?

I have data consisting of the distance cats travelled each hour, collected over four or five days per cat. The cats were measured in different seasons and I want to know whether the activity of the cat varied with hour, dependent on the season. For example, were cats more nocturnal in summer? I am not sure whether the output is showing that there is a difference in the mean activity, or a difference in the pattern of activity over time?

My data is structured with the headings: Cat_ID, season, day, hour, distance.

Using the mgvc package, my GAMM model is then constructed like this:

mod1 <- gamm(distance ~ s(hour) + s(hour,by=season,bs="cc",fx=TRUE,k=6) + season,
random = list(id = ~1), data = dn, na.action=na.exclude)


The output is:

> summary(mod1$gam) Family: gaussian Link function: identity Formula: dn$distance ~ s(hour) + s(hour, by = season, bs = "cc", fx = TRUE,
k = 6) + season

Parametric coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)    160.77      13.68  11.756   <2e-16 ***
seasonSummer   -15.33      18.33  -0.836    0.403
seasonWinter   -25.49      17.74  -1.437    0.151
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Approximate significance of smooth terms:
edf Ref.df     F  p-value
s(hour)                1      1 5.526 0.018769 *
s(hour):seasonSpring   4      4 5.009 0.000499 ***
s(hour):seasonSummer   4      4 1.529 0.190775
s(hour):seasonWinter   4      4 2.856 0.022273 *
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

1: In as.numeric(object$$y) - object$$fitted.values :
2: In w * (as.numeric(object$$y) - object$$fitted.values) :