Correlation of model and experimental results I am trying to find a scientifically accepted method to compare my model predictions with experimental values. Here, my model predicts the concentration of DNA as a function of time. DNA concentration is also measured at several points in time in vivo in an experimental setting. 
Is it valid to compare the model predictions to observed values with a Pearson Correlation Coefficient? Would a considerably big coefficient be taken as an accepted 'method' to prove that model predictions are close to experimental and, thus, that the model is valid?
 A: In medical sciences, calibration can refer to the precision of assays. The goal of your comparison is a calibration study to determine if your model is valid for predicting DNA concentration as a function of time. 
The basic concept of a Pearson correlation coefficient is sufficient to identify discrimination: that is that comparing any two observations, better discrimination means that the one with higher predicted values is likely to have higher observed values. This is a result of the fact a correlation coefficient is a weighted average of the pairwise slopes between all observed values.
A negative consequence of the Pearson correlation coefficient is that it is invariant to scale/shift. That means that with an observed sequence [0,1,2,3,4] and predicted sequence [500,505,510,515,520] your agreement is 100% but the predictions are orders of magnitude off.
If you want a measure of predictive accuracy that ranges from -1 to 1 and achieves a value of 1 if and only if the predictions are identical to observed values, fit the following linear model:
$$ E[\text{Observed}_i|\text{Predicted}_i] = 0 + 1 \text{Predicted}_i$$ 
This can be done with regression through the origin and fitting the predicted value as an offset term. Then, using model statistics, calculate the square root of the $R^2$ coefficient from that. The sign of direction must be found post-hoc, but let us assume it's not an issue unless the model in question is terrible. 
This model has a correspondence with the concordance calibration coefficient of Lin 1989. There are analogues for mixed/repeated measures designs.
An example of performing such a design is shown here. I am using data from Margaret Pepe's study on inspecting a biomarker's relation to development of AKI (Acute Kidney Injury) following cardiac surgery. The biomarker is longitudinal, but I am using just the first observed value for each patient.
aki <- read.csv('http://research.fhcrc.org/content/dam/stripe/diagnostic-biomarkers-statistical-center/files/aki_sim.csv')
aki <- aki[!duplicated(aki$id),]
set.seed(123)
itrain <- sample(1:nrow(aki), nrow(aki)/2)
aki.train <- aki[itrain, ]
aki.test <- aki[-itrain, ]
pred.model <- lm( statusall ~ y, data=aki.train)
pred <- predict(pred.model, newdata=aki.test)

plot(pred, aki.test$statusall, xlab='Predicted status', 
  ylab='Observed status \n(0: no AKI, 1: mild AKI, 2: severe AKI, 9: death)')
abline(0,1)
calib.model <- lm(statusall ~ 0 + offset(pred), data=aki.test)

Gives:


summary(calib.model)

Call:
lm(formula = statusall ~ 0 + offset(pred), data = aki.test)

Residuals:
    Min      1Q  Median      3Q     Max 
-0.6599 -0.4257 -0.3525 -0.2539  8.7043 

No Coefficients

Residual standard error: 0.9518 on 900 degrees of freedom

So $\sqrt{1-0.9518}$ gives a calibration coefficient of 0.22 which is very poor. This is likely a result of the ordinal encoding of AKI severity which includes 9 as death from AKI. Obviously, biological measurements like creatinine clearance would be preferable. 
Excluding "9"s as deaths: the R^2 is much better:
> calib.model <- lm(statusall ~ 0 + offset(pred), data=aki.test, subset=statusall != 9)
> summary(calib.model)

Call:
lm(formula = statusall ~ 0 + offset(pred), data = aki.test, subset = statusall != 
    9)

Residuals:
    Min      1Q  Median      3Q     Max 
-0.6599 -0.4269 -0.3528 -0.2570  1.8659 

No Coefficients

Residual standard error: 0.6419 on 894 degrees of freedom

Gives a calibration coefficient of sqrt(1-0.6419) = 0.60 which is still low. 
A last issue to consider is: what is the appropriate scale on which to calibrate the expected and observed values? The raw scale is nice because it is interpretable, but DNA concentrations are often measured on a log scale or a complementary log-log scale. Consider these changes of variable based on your knowledge of the problem.
