Graphing Bean Sprout Respiration with the Hadleyverse

Tidying Data

I imported the data and gathered them into long form.

sprouts <- read.csv("../datasets/respiration.csv")
sprouts <- tbl_df(sprouts)
sprouts <- gather(sprouts, "minutes", "distance", 4:7)
sprouts$minutes <- recode(sprouts$minutes, "X5.min" = 5, "X10.min" = 10,
       "X15.min" = 15, "X20.min" = 20)
sprouts

## # A tibble: 680 × 6
##    Teacher Period  Table    Age minutes distance
##     <fctr>  <int> <fctr> <fctr>   <dbl>    <dbl>
## 1    Ching      1      a  Young       5       18
## 2    Ching      1      b  Young       5      150
## 3    Ching      1      c  Young       5       85
## 4    Ching      1      d  Young       5      114
## 5    Ching      1      e  Young       5       NA
## 6    Ching      2      a  Young       5       88
## 7    Ching      2      b  Young       5       59
## 8    Ching      2      c  Young       5       71
## 9    Ching      2      d  Young       5       69
## 10   Ching      2      e  Young       5       NA
## # ... with 670 more rows

Calculating Summary Statistics

I calculated means and standard errors by time and age.

plot_data <- sprouts %>% group_by(minutes, Age) %>% 
  summarize(avg_dist = mean(distance, na.rm = T), 
            se_dist = sd(distance, na.rm = T)/sqrt(sum(!is.na(distance))))

plot_data

## Source: local data frame [8 x 4]
## Groups: minutes [?]
## 
##   minutes    Age  avg_dist   se_dist
##     <dbl> <fctr>     <dbl>     <dbl>
## 1       5    Old  38.51948  2.524241
## 2       5  Young  73.00000  5.172839
## 3      10    Old  74.76923  4.802707
## 4      10  Young 140.24324  8.619536
## 5      15    Old 109.75641  6.897758
## 6      15  Young 201.18919 12.290029
## 7      20    Old 143.44595  9.555449
## 8      20  Young 255.94286 14.874994

Data Visualization

I plotted the mean and standard errors using ggplot2.

pd <- position_dodge(0)
ggplot(plot_data, aes(x=minutes, y = avg_dist, color = Age, group = Age)) +
  geom_errorbar(aes(ymin=avg_dist-se_dist, ymax=avg_dist+se_dist), 
                width=.1, position=pd) + 
  geom_point(position=pd) +
  ylab("distance (mm)") +
  ggtitle("Mean (s.e.) Respiration of Young and Old Bean Sprouts") +
  expand_limits(x = 0, y = 0) # to show the origin 

The result was a strikingly linear relationship between distance and time for both age groups.

Model of Age Effect on Respiration Rate

To fit a model to these, I used linear regression. I modeled this using an interaction of time and age.

model1 <- lm(distance ~ minutes + Age + minutes * Age, data = sprouts)
summary(model1)

## 
## Call:
## lm(formula = distance ~ minutes + Age + minutes * Age, data = sprouts)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -182.183  -32.157   -8.183   20.874  216.890 
## 
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)    
## (Intercept)        4.1723     8.2336   0.507    0.613    
## minutes            6.9969     0.6049  11.567  < 2e-16 ***
## AgeYoung          10.8904    14.4759   0.752    0.452    
## minutes:AgeYoung   5.2091     1.0648   4.892 1.39e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 58.86 on 449 degrees of freedom
##   (227 observations deleted due to missingness)
## Multiple R-squared:  0.5216, Adjusted R-squared:  0.5184 
## F-statistic: 163.2 on 3 and 449 DF,  p-value: < 2.2e-16

The model came out pretty nice! Here it is in algebraic form where \(a\) is age (young corresponds to a = 1 and old corresponds to a = 0) and \(t\) is time elapsed in minutes:

\[distance = 4.2 + 7t + 10.9a + 5.2ta\]

When Age = 0 (old sprouts), the equation changes to:

\[distance_{old \space sprouts} = 4.2 + 7t \]

When Age = 1 (young sprouts), the equation changes to:

\[distance_{young \space sprouts} = 15.1 + 12.2t\]

The model only explained about 50% of the variance as seen when we overlay the regression lines over the raw data.

ggplot(sprouts, aes(x=minutes, y=distance, color=Age)) +
  geom_point(shape = 1, position=position_jitter(width=.5,height=0)) +
  geom_smooth(method=lm,   # Add linear regression lines
                se=FALSE) +
  ggtitle("Respiration of Young and Old Bean Sprouts")

## Warning: Removed 227 rows containing non-finite values (stat_smooth).

## Warning: Removed 227 rows containing missing values (geom_point).