DOY<-c(102,102,102,102,102,102,102,102,102,102,212,212,212,212,212,212, 212,212,212,212)
LOCATION <- c(1,1,1,1,1,2,2,2,2,2,1,1,1,1,1,3,3,3,3,3)
response <-c(NA,NA,NA,NA,NA,7,10,15,20,30,2,4,6,NA,8,10,15,20,30,NA) 
ts <- c(0,10,20,30, 40,0,10,20,30,40,0,10,20,30,40,0,10,20,30,40)
test.data <- data.frame(cbind(DOY, LOCATION, response, ts))


library(tidyverse)
library(broom)
pos1 <-  possibly(lm, otherwise = NULL)
prsq <- possibly(pull, otherwise = NA)
test.data %>%
     group_by(DOY, LOCATION) %>%
     nest %>%
     mutate(model = map(data, ~ pos1(response~ ts, data = .x)),
            slope = map_dbl(model, ~ 
                            .x %>% 
                                tidy %>%
                                filter(term == 'ts') %>%
                                prsq(estimate)),
            R2 = map_dbl(model, ~ 
                             .x %>%
                                   glance %>%
                                   prsq(r.squared))) %>%
      select(-data, -model)
# A tibble: 4 x 4
#    DOY LOCATION  slope     R2
#          
#1   102        1 NA     NA    
#2   102        2  0.56   0.953
#3   212        1  0.149  0.966
#4   212        3  0.650  0.966