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I was wondering if someone could help me understand what the parameter estimates and p-values are saying in a three-parameter exponential decay function using the drm function in the 'drc' package in R. Using the help guide, the line equation should be: f(x) = c + (d - c)(exp(-x/e)), where c is the lower limit (asymptote), d is the upper limit or origin, and e is the rate constant.

I have the following example dataset of glycogen measurements in starving tick larvae over time:

      Assay TickAge Measurement_in_ug PerTick Sample
1  Glycogen       0              5.92    0.59  2.1.1
2  Glycogen       0              6.35    0.64  2.1.2
3  Glycogen       0              7.82    0.78  2.1.3
4  Glycogen       2              5.20    0.52  2.2.1
5  Glycogen       2              4.07    0.41  2.2.2
6  Glycogen       2              4.85    0.48  2.2.3
7  Glycogen       4              3.87    0.39  2.3.1
8  Glycogen       4              2.92    0.29  2.3.2
9  Glycogen       4              4.27    0.43  2.3.3
10 Glycogen       6              3.81    0.38  2.4.1
11 Glycogen       6              3.09    0.31  2.4.2
12 Glycogen       6              4.36    0.44  2.4.3
13 Glycogen       8              3.41    0.34  2.5.1
14 Glycogen       8              2.14    0.21  2.5.2
15 Glycogen       8              2.54    0.25  2.5.3

with this model:

Glyc3 = drm(PerTick~TickAge, fct = EXD.3(), data=Glyc)

And this output:

summary(Glyc3)

Model fitted: Shifted exponential decay (3 parms)

Parameter estimates:

              Estimate Std. Error t-value  p-value    
c:(Intercept) 0.262314   0.084366  3.1092 0.009034 **
d:(Intercept) 0.665572   0.042465 15.6733 2.35e-09 ***
e:(Intercept) 3.282096   1.814096  1.8092 0.095520 .`  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error:
0.07326667 (12 degrees of freedom)

 

Specifically, I am interested how to interpret the non-significant e. I know that it is a Wald test with H0: e = 0. How does this change the exponential decay equation mentioned above?

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1 Answer 1

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Nonsignificant coefficient means either 1) the coefficient is not significantly different from zero (i.e. there is no exponential decay, H0: e=0) or 2) you have not determined the coefficient with reasonable certainty (possibly need more/better data). At face value, such insignificance suggests that unless you establish a significant rate constant, you would do as good by predicting the outcome to be the mean glycogen level across all your measurements.

grand_mean <- mean(Glyc$PerTick)
plot(Glyc3, type='all')
abline(h=grand_mean, lty=2)
text(0.3, 0.45, paste('grand mean =', signif(grand_mean, 1)))

per tick

You can, of course, apply domain knowledge and use your rate constant no matter how poorly determined. Looking at your data, you probably do have a decay, it is just that you cannot be confident enough about your rate estimate. Indeed:

print(confint(Glyc3), digits=2)
#                2.5 % 97.5 %
# c:(Intercept)  0.078   0.45
# d:(Intercept)  0.573   0.76
# e:(Intercept) -0.670   7.23

The higher end of your estimate is very different from zero (thus high rate cannot be ruled out) yet confidence intervals span over one order of magnitude, from -0.67 to 7 (including zero, thus termed insignificant). You could improve by acquiring more data, of particular interest would be TickAge < 2 and perhaps TickAge > 8.

Finally, regarding the equation: f(x) = c + (d - c)(exp(-x/e)). If e=0, then exp(-x/e)=-Inf. That is exp(-x/e)=0 for all practical purposes and then you have f(x) = c. You are left with a single coefficient c. Since there is no decay (at least it cannot be described with a reasonable confidence just based on data), you estimate the c by taking mean across all your values.

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