Correlation/Regression

Distance Score and Alzheimer’s Score

dat <- tribble(~participant, ~distance, ~alzheimers,
               1, 2, 7, 
               2, 8, 20, 
               3, 10, 23, 
               4, 8, 19, 
               5, 4, 15,
               6, 7, 14, 
               7, 5, 10,
               8, 4, 12)

Scatterplot

dat %>% 
  ggplot(aes(x = distance, y = alzheimers)) +
  geom_point() +
  geom_smooth() +
  theme_classic() +
  labs(title = "Peanut Butter Smell Test & Alzheimer's Screening Scores",
       x = "Distance",
       y = "Alzheimer's Screening Score")

## `geom_smooth()` using method = 'loess' and formula = 'y ~ x'

Descriptives

dat %>% 
  summarise(x_mean = mean(distance),
            x_sd = round(sd(distance), 2),
            y_mean = mean(alzheimers),
            y_sd = round(sd(alzheimers), 2))

## # A tibble: 1 × 4
##   x_mean  x_sd y_mean  y_sd
##    <dbl> <dbl>  <dbl> <dbl>
## 1      6  2.67     15   5.4

Assumptions

#linearity 
model <- lm(distance ~ alzheimers, data = dat)

plot(lm(distance ~ alzheimers, data = dat), which = 1)

#normality
plot(lm(distance ~ alzheimers, data = dat), which = 2)

Correlation

corr <- correlation(data = dat,
                    select = "distance",
                    select2 = "alzheimers",
                    method = "Pearson",
                    alternative = "two.sided")

corr

## # Correlation Matrix (Pearson-method)
## 
## Parameter1 | Parameter2 |    r |       95% CI | t(6) |       p
## --------------------------------------------------------------
## distance   | alzheimers | 0.91 | [0.58, 0.98] | 5.41 | 0.002**
## 
## p-value adjustment method: Holm (1979)
## Observations: 8

APA Table

dat %>% 
  select(-participant) %>% 
  apa.cor.table(filename = "assignment2_correlation.doc")

## 
## 
## Means, standard deviations, and correlations with confidence intervals
##  
## 
##   Variable      M     SD   1         
##   1. distance   6.00  2.67           
##                                      
##   2. alzheimers 15.00 5.40 .91**     
##                            [.58, .98]
##                                      
## 
## Note. M and SD are used to represent mean and standard deviation, respectively.
## Values in square brackets indicate the 95% confidence interval.
## The confidence interval is a plausible range of population correlations 
## that could have caused the sample correlation (Cumming, 2014).
##  * indicates p < .05. ** indicates p < .01.
## 

summary(model)

## 
## Call:
## lm(formula = distance ~ alzheimers, data = dat)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.00000 -0.45588 -0.02941  0.60784  1.45098 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)   
## (Intercept) -0.76471    1.31969  -0.579  0.58337   
## alzheimers   0.45098    0.08338   5.409  0.00165 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.191 on 6 degrees of freedom
## Multiple R-squared:  0.8298, Adjusted R-squared:  0.8014 
## F-statistic: 29.25 on 1 and 6 DF,  p-value: 0.00165

library(report)
report(model)

## We fitted a linear model (estimated using OLS) to predict distance with
## alzheimers (formula: distance ~ alzheimers). The model explains a statistically
## significant and substantial proportion of variance (R2 = 0.83, F(1, 6) = 29.25,
## p = 0.002, adj. R2 = 0.80). The model's intercept, corresponding to alzheimers
## = 0, is at -0.76 (95% CI [-3.99, 2.46], t(6) = -0.58, p = 0.583). Within this
## model:
## 
##   - The effect of alzheimers is statistically significant and positive (beta =
## 0.45, 95% CI [0.25, 0.66], t(6) = 5.41, p = 0.002; Std. beta = 0.91, 95% CI
## [0.50, 1.32])
## 
## Standardized parameters were obtained by fitting the model on a standardized
## version of the dataset. 95% Confidence Intervals (CIs) and p-values were
## computed using a Wald t-distribution approximation.

Regression

model <- lm(distance ~ alzheimers, data = dat)

plot(model, which = 1)

plot(model, which = 2)

plot(model, which = 3)

summary(model)

## 
## Call:
## lm(formula = distance ~ alzheimers, data = dat)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.00000 -0.45588 -0.02941  0.60784  1.45098 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)   
## (Intercept) -0.76471    1.31969  -0.579  0.58337   
## alzheimers   0.45098    0.08338   5.409  0.00165 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.191 on 6 degrees of freedom
## Multiple R-squared:  0.8298, Adjusted R-squared:  0.8014 
## F-statistic: 29.25 on 1 and 6 DF,  p-value: 0.00165

report(model)

## We fitted a linear model (estimated using OLS) to predict distance with
## alzheimers (formula: distance ~ alzheimers). The model explains a statistically
## significant and substantial proportion of variance (R2 = 0.83, F(1, 6) = 29.25,
## p = 0.002, adj. R2 = 0.80). The model's intercept, corresponding to alzheimers
## = 0, is at -0.76 (95% CI [-3.99, 2.46], t(6) = -0.58, p = 0.583). Within this
## model:
## 
##   - The effect of alzheimers is statistically significant and positive (beta =
## 0.45, 95% CI [0.25, 0.66], t(6) = 5.41, p = 0.002; Std. beta = 0.91, 95% CI
## [0.50, 1.32])
## 
## Standardized parameters were obtained by fitting the model on a standardized
## version of the dataset. 95% Confidence Intervals (CIs) and p-values were
## computed using a Wald t-distribution approximation.