Paired t-test with random effects? I have a dataset where we sampled the number of aphids on twelve "pairs" of plants in each of 6 different fields. The pairs in each field represented a plant infected with a pathogen, and an adjacent uninfected plant (< 10 cm away). Thus, there were a total of 72 "pairs" across the 6 fields. These pairs were separated by over 1000m and I want to treat each pair as an independent experimental unit. I had originally planned to just do a paired t-test with all the pairs as follows:
t.test(Uninfected, Infected, paired=T)

But, I worry that I should also be including a random effect of field. Is there a way to code this so I have a random effect of field but can still determine if the aphid densities on the two types of plants differ significantly? And if so, do I need to reformat my data? Right now it's set up with the number of aphids on uninfected plants in one column, and the number of aphids on infected plants in a separate column, where the rows represent the unique pairs.
 A: Given the experiment you described, I would opt for a generalized linear mixed-effects model using a Poisson distribution (since you have count data) and FIELD and PAIR as random effects. The random effects allow you to capture and account for the variation in PAIR and FIELD.
To illustrate what I mean, I built an example dataset:
set.seed(123)
df <- data.frame(PAIR=rep(c(1:72), each=2),
                 TREAT=rep(c("infected","control"), 72),
                 FIELD=rep(c(1:6),each=24),
                 COUNT=sample(c(0:100),144, replace = T))

Then fitting the GLMM using the glmer() function from the lme4 package: 
require(lme4)
f1 <- glmer(COUNT ~ TREAT + (1|PAIR) + (1|FIELD), family = poisson, data = df)

Check the model output:
summary(f1)        
Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
 Family: poisson  ( log )
Formula: COUNT ~ TREAT + (1 | PAIR) + (1 | FIELD)
   Data: df

     AIC      BIC   logLik deviance df.resid 
  2537.3   2549.2  -1264.7   2529.3      140 

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-6.4254 -2.2944 -0.0113  2.2976  6.4707 

Random effects:
 Groups Name        Variance  Std.Dev. 
 PAIR   (Intercept) 2.085e-01 4.566e-01
 FIELD  (Intercept) 2.913e-09 5.398e-05
Number of obs: 144, groups:  PAIR, 72; FIELD, 6

Fixed effects:
               Estimate Std. Error z value Pr(>|z|)    
(Intercept)    3.840378   0.056563   67.90   <2e-16 ***
TREATinfected -0.003276   0.023309   -0.14    0.888    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Correlation of Fixed Effects:
            (Intr)
TREATinfctd -0.206

Here you can check the random and fixed effect output as well as other information. For example, the line Number of obs: 144, groups:  ID, 72; FIELD, 6 allows you to check whether the specified groupings in the random statement make sense. Depending on how your data is coded, you might have to adjust the random statement to reflect the intended groupings of the experiment.
Now we moved from a simple paired t-test to a rather complicated GLMM - but I think given what you described above, this seems reasonable.
Regarding model specification also have a look here. For more examples on GLMMs, have a look here as well. There is also a lot of information here on Cross Validated regarding this topic.
A: I agree with Stephan that you should be using a count distribution (like Poisson or negative binomial), but since your have so few fields and you're not interested in the field effects, you can simply do a regression comparison and use standard errors that are cluster-robust. I see you're using R, but in Stata this would be straightforward to implement. You might create a row for each observation, and a colmn for its "pair-field" combination, the use
nbreg num_aphids infected, vce(cluster pair_field)

