Example 02: Hypothesis Testing

Research Question & Data

Context:

• Data were collected for two species: O. exclamationis and O. niveus. The data are contained in a data frame called crickets with a total of 31 data points.

• Study on cricket chirp rates and temperature for two species

• Data:

  • Temperature (°C)

  • Chirps per minute

  • Species: (A) O. exclamationis (B) O. niveus

Motivation:

• Scientists are interested in whether the temperature affects crickets’ chirp rates.

Research Questions:

• RQ1: Does temperature affect chirp rate in general?

• RQ2: Do species differ in their chirp rates?

• RQ3: Do species respond differently to temperature?

Load the data into the R

## If you did not install modeldata and tidyverse packages, execute following codes:
install.packages("modeldata")
install.packages("tidyverse")

library(tidyverse)
data(crickets, package = "modeldata")

Data Visualization

ggplot(crickets, 
       aes(x = rate, fill = species, color = species)) + 
  geom_density(alpha = .5)

ggplot(crickets, 
       aes(x = temp, y = rate, color = species)) + 
  geom_point() +
  geom_smooth(method = lm, se = FALSE) +
  labs(x = "Temperature (C)", 
       y = "Chirp Rate (per minute)")

Statistical Hypotheses & Model

Null Hypotheses:

1. Temperature has no effect on the chirp rate.
2. There are no differences between the species’ chirp rate,
   • e.g., overall chirp rate and sensitivity to temperature

Null Hypotheses for hypothesis testing:

1. H₀: Temperature has no effect (β_{temp} = 0)

2. H₀: No species difference (β_{species} = 0)

3. H₀: No interaction (β_{temp×species} = 0)

# Full model with interaction
interaction_fit <- lm(rate ~ (temp + species)^2, data = crickets)

Model Diagnostics

par(mfrow = c(1, 2))
plot(interaction_fit, which = 1) # Residuals vs Fitted
plot(interaction_fit, which = 2) # Normal Q-Q

Testing Interaction Effect

# Fit main effects model
main_effect_fit <- lm(rate ~ temp + species, data = crickets)

# Compare models
anova(main_effect_fit, interaction_fit)

Analysis of Variance Table

Model 1: rate ~ temp + species
Model 2: rate ~ (temp + species)^2
  Res.Df    RSS Df Sum of Sq     F Pr(>F)
1     28 89.350                          
2     27 85.074  1    4.2758 1.357 0.2542

Result: Interaction not significant (p > 0.05). Choose simpler model without interaction

Final Model Results

summary(main_effect_fit)

Call:
lm(formula = rate ~ temp + species, data = crickets)

Residuals:
    Min      1Q  Median      3Q     Max 
-3.0128 -1.1296 -0.3912  0.9650  3.7800 

Coefficients:
                  Estimate Std. Error t value Pr(>|t|)    
(Intercept)       -7.21091    2.55094  -2.827  0.00858 ** 
temp               3.60275    0.09729  37.032  < 2e-16 ***
speciesO. niveus -10.06529    0.73526 -13.689 6.27e-14 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.786 on 28 degrees of freedom
Multiple R-squared:  0.9896,    Adjusted R-squared:  0.9888 
F-statistic:  1331 on 2 and 28 DF,  p-value: < 2.2e-16

Interpretation:

• Temperature effect: +3.75 chirps/°C (p < 0.001)

• O. niveus: 17 fewer chirps/minute (p < 0.001)

Model Predictions

new_temps <- data.frame(
  species = "O. exclamationis",
  temp = seq(15, 25, by = 5)
)

predict(main_effect_fit, new_temps)

       1        2        3 
46.83039 64.84415 82.85792 

Conclusions

Statistical Evidence:

1. Strong temperature effect (p < 0.001)

2. Clear species difference (p < 0.001)

3. Similar temperature response (no interaction)

Limitations:

• Valid within observed temperature range (15-32°C)

• Only two species studied

Takeaway Note: the key steps

Here are the key steps for performing inferential statistics using the cricket example:

1. Data visualization and propose Null Hypotheses
   • Temperature has no effect on chirp rate
   • No differences between species’ chirp rates
2. Fit Initial Model
   • Used lm() with interaction terms between temperature and species
   • Formula: rate ~ (temp + species)^2
3. Check Model Diagnostics
   • Examined residual plots vs predicted values
   • Checked normality of residuals using Q-Q plot
   • Verified assumptions were reasonable
4. Test Model Components
   • Compared models with/without interaction using ANOVA
   • Found interaction term was not statistically significant (p > 0.05)
   • Selected simpler main effects model
5. Interpret Final Model
   • Analyzed coefficients and p-values for temperature and species effects
   • Temperature showed significant positive effect
   • Species showed significant difference in baseline chirp rates
   • Limited conclusions to observed temperature range