The effect size of difference

Question

I have this interesting data where I would like to estimate possibly a parameter of the difference (between $A+B$ and $A+C$, inference using both) that would allow me to infer the development of $A$ (whether there is a propensity to decrease or increase).

Any hint as to how to approach it included type of modeling/estimation procedure?

Here is part of the data: The data itself is a rate of observing number of species in days.

These have been calculated in R based on this formula for $A$:

A = obs / mean(obs.window)

The values of $B$ and $C$ in R are based on the formulas:

B = obs / min(obs.window)

and

C = obs / max(obs.window)

where obs is a observed number of species per day and obs.window is a average value of a sliding window of $10$ days (moving average).

 x <- "A B C 
 1  0.63 0.67 0.61
 2  0.62 0.64 0.60
 3  0.64 0.65 0.59
 4  0.70 0.70 0.63
 5  0.71 0.73 0.68
 6  0.70 0.75 0.69
 7  0.71 0.75 0.70
 8  0.74 0.76 0.71
 9  0.79 0.81 0.74
10 0.80 0.83 0.76
11 0.82 0.84 0.78
12 0.82 0.84 0.80
13 0.83 0.85 0.81
14 0.81 0.88 0.80
15 0.78 0.84 0.77
16 0.75 0.79 0.74
17 0.73 0.77 0.72
18 0.72 0.75 0.71
19 0.73 0.75 0.71
20 0.73 0.75 0.71
21 0.74 0.76 0.72
22 0.72 0.76 0.71
23 0.71 0.74 0.69
24 0.73 0.75 0.70
25 0.78 0.79 0.71
26 0.82 0.84 0.77
27 0.80 0.84 0.78
28 0.77 0.81 0.76
29 0.79 0.81 0.75
30 0.83 0.84 0.78
31 0.86 0.87 0.82
32 0.85 0.87 0.83
33 0.83 0.84 0.82
34 0.78 0.85 0.77
35 0.74 0.80 0.72
36 0.72 0.76 0.71
37 0.74 0.77 0.70
38 0.75 0.75 0.70
39 0.78 0.81 0.72
40 0.78 0.82 0.75" 

And here some adjustment:

data <- read.table(text=x, header = TRUE)

data$diff_AC <- with(data, (A-C))
data$diff_AB <- with(data, (A-B))

with(data, plot(A~1, col=1))
with(data, points(B~1, col=2))
with(data, points(C~1, col=3))

EDIT: I'm interested in estimation of the relationship of the interval and its overall with A as an idicator.

However, I was thinking using Beta distribution and simulation on rolling window of 20 days, would this be anyhing meaningful?

windw <- 20; # rolling windows size

for(i in 1:10){
    dt <- data[(1:i):(i+windw), ] 
    # mean & variance
    dt$mu_A <- with(dt, mean(A))
    dt$sig_A <- with(dt, var(A))
    # estimate alpah & beta via moments
    moment_alpha_A <- with(dt, mu_A*(mu_A*(1-mu_A)/sig_A-1))
    moment_beta_A <- with(dt, (1-mu_A)*(mu_A*(1-mu_A)/sig_A-1))
    # simulate
    rb_A <- rbeta(10000, unique(moment_alpha_A), unique(moment_beta_A))
    # plot
    with(dt, plot(A~1, col=1))
    abline(h=median(rb_A), col="blue3", lwd=3)
    abline(h=tail(dt$A,1), col="red3", lwd=1)
    Sys.sleep(1)
}

Answer 1

I'm not sure but here is the best solution I can provide:

I feel sort of optimization should be used to solve this issue, or definitely better model (rather then linear OLS model) but nonlinear...

data$retA <- with(data, as.numeric(c(0,diff(A))/lag(A,1)))


diff_binAB <- with(data, unique(diff_AB))
mse <- numeric(length(diff_binAB))

for(i in 1:length(diff_binAB)){
    pwise <- with(data, lm(retA ~ diff_AB*(diff_AB < diff_binAB[i]) + diff_AB*(diff_AB >= diff_binAB[i])))
    mse[i] <- summary(pwise)[6]
   }

mse <- as.numeric(mse)
mse 

diff_binAB[which(mse==min(mse))]
# -0.07

diff_binAC <- with(data, unique(diff_AC))
mse1 <- numeric(length(diff_binAC))

for(i in 1:length(diff_binAC)){

    pwise <- with(data, lm(retA ~ diff_AC*(diff_AC < diff_binAC[i]) + diff_AC*(diff_AB >= diff_binAC[i])))
    mse1[i] <- summary(pwise)[6]
   }

mse1 <- as.numeric(mse1)
mse1 

diff_binAC[which(mse1==min(mse1))]
# 0.04 

Here the results would suggest that the return (rate of change) is explained if the difference between A and B is at -0.07 (negative difference) and 0.04 with possitive difference between A and C.