Reliability Estimates

KR20 Estimate

library(dplyr)
library(tidyverse)
data <- read.csv("/cloud/project/Reliability Assignment.csv")

#number of items is denoted by n
(n <- ncol(data))

## [1] 25

#number of students is denoted by m
(m <- nrow(data))

## [1] 20

proportion of sample who answered item correctly (p)

p <- apply(data,2,mean)
p

##  sit1  sit2  sit3  sit4  sit5  sit6  sit7  sit8  sit9 sit10 sit11 sit12 sit13 
##  0.95  0.70  0.95  0.85  0.85  0.75  0.80  0.65  0.60  0.80  0.70  0.50  0.70 
## sit14 sit15 sit16 sit17 sit18 sit19 sit20 sit21 sit22 sit23 sit24 sit25 
##  0.40  0.60  0.70  0.45  0.45  0.50  0.45  0.45  0.45  0.30  0.40  0.60

proportion of sample who answered item incorrectly (q)

q <- 1-p
q

##  sit1  sit2  sit3  sit4  sit5  sit6  sit7  sit8  sit9 sit10 sit11 sit12 sit13 
##  0.05  0.30  0.05  0.15  0.15  0.25  0.20  0.35  0.40  0.20  0.30  0.50  0.30 
## sit14 sit15 sit16 sit17 sit18 sit19 sit20 sit21 sit22 sit23 sit24 sit25 
##  0.60  0.40  0.30  0.55  0.55  0.50  0.55  0.55  0.55  0.70  0.60  0.40

sum of pq

sum_pq <- p*q
sum_pq

##   sit1   sit2   sit3   sit4   sit5   sit6   sit7   sit8   sit9  sit10  sit11 
## 0.0475 0.2100 0.0475 0.1275 0.1275 0.1875 0.1600 0.2275 0.2400 0.1600 0.2100 
##  sit12  sit13  sit14  sit15  sit16  sit17  sit18  sit19  sit20  sit21  sit22 
## 0.2500 0.2100 0.2400 0.2400 0.2100 0.2475 0.2475 0.2500 0.2475 0.2475 0.2475 
##  sit23  sit24  sit25 
## 0.2100 0.2400 0.2400

Grand sum of pq:\(\sum{pq}\)

(Grand_sum <- sum(sum_pq))

## [1] 5.0725

Total score by test taker

Total_scores <- rowSums(data)
test_takers <- c(paste("Student",1:m))
as.data.frame(Total_scores, row.names = test_takers)

##            Total_scores
## Student 1            11
## Student 2            13
## Student 3            21
## Student 4             9
## Student 5            13
## Student 6            12
## Student 7            25
## Student 8            23
## Student 9            25
## Student 10           10
## Student 11           17
## Student 12           11
## Student 13           12
## Student 14           14
## Student 15           18
## Student 16           20
## Student 17            7
## Student 18           22
## Student 19            7
## Student 20           21

Variance of total scores by students

Variance <- var(Total_scores)
(Pop_var <- ((m-1)/m)*Variance)

## [1] 33.2475

KR20

KR20 <- (n / (n - 1)) * (1 - (Grand_sum/Pop_var))
KR20

## [1] 0.8827418

Split half: Odd-Even Items

Odd items

item_odd <- data[, c(TRUE,FALSE)]
head(item_odd)

##   sit1 sit3 sit5 sit7 sit9 sit11 sit13 sit15 sit17 sit19 sit21 sit23 sit25
## 1    1    1    0    0    1     0     1     1     0     0     1     0     1
## 2    1    1    1    1    1     1     1     0     0     0     0     0     0
## 3    1    1    1    1    0     1     1     1     1     1     1     0     1
## 4    1    1    1    0    0     1     0     0     0     1     0     0     0
## 5    1    1    1    1    1     1     1     0     0     0     0     0     0
## 6    1    1    1    1    0     1     0     0     0     1     0     0     0

Even items

item_even <- data[, c(FALSE,TRUE)]
head(item_even)

##   sit2 sit4 sit6 sit8 sit10 sit12 sit14 sit16 sit18 sit20 sit22 sit24
## 1    0    1    0    0     1     1     0     0     1     0     0     0
## 2    1    1    1    0     1     1     1     0     0     0     0     0
## 3    1    1    1    0     1     1     1     1     1     1     0     1
## 4    0    1    0    0     1     0     0     1     0     1     0     0
## 5    1    1    1    1     1     1     0     0     0     0     0     0
## 6    1    1    1    0     0     0     0     1     1     0     0     1

Total score by test takers by test subset

test_takers <- c(paste("Student",1:m))
Total_Odd <- rowSums(item_odd)
Total_Even <- rowSums(item_even)
Split_Total <- data.frame(Total_Odd,Total_Even)
as.data.frame(Split_Total, row.names = test_takers)

##            Total_Odd Total_Even
## Student 1          7          4
## Student 2          7          6
## Student 3         11         10
## Student 4          5          4
## Student 5          7          6
## Student 6          6          6
## Student 7         13         12
## Student 8         13         10
## Student 9         13         12
## Student 10         5          5
## Student 11         9          8
## Student 12         5          6
## Student 13         8          4
## Student 14         9          5
## Student 15        10          8
## Student 16        12          8
## Student 17         2          5
## Student 18        11         11
## Student 19         4          3
## Student 20        12          9

Correlation for the two subsets

(cor_odd_even <- cor(Total_Odd, Total_Even))

## [1] 0.8459988

Split half first and last half Items

First half items

item_first <- data[, c(1:ceiling((n/2)))]
head(item_first)

##   sit1 sit2 sit3 sit4 sit5 sit6 sit7 sit8 sit9 sit10 sit11 sit12 sit13
## 1    1    0    1    1    0    0    0    0    1     1     0     1     1
## 2    1    1    1    1    1    1    1    0    1     1     1     1     1
## 3    1    1    1    1    1    1    1    0    0     1     1     1     1
## 4    1    0    1    1    1    0    0    0    0     1     1     0     0
## 5    1    1    1    1    1    1    1    1    1     1     1     1     1
## 6    1    1    1    1    1    1    1    0    0     0     1     0     0

Second half items

item_second <- data[, c(ceiling((n/2)+1):n)]
head(item_second)

##   sit14 sit15 sit16 sit17 sit18 sit19 sit20 sit21 sit22 sit23 sit24 sit25
## 1     0     1     0     0     1     0     0     1     0     0     0     1
## 2     1     0     0     0     0     0     0     0     0     0     0     0
## 3     1     1     1     1     1     1     1     1     0     0     1     1
## 4     0     0     1     0     0     1     1     0     0     0     0     0
## 5     0     0     0     0     0     0     0     0     0     0     0     0
## 6     0     0     1     0     1     1     0     0     0     0     1     0

Total score by test takers by test subset

test_takers <- c(paste("Student",1:m))
Total_First <- rowSums(item_first)
Total_Second <- rowSums(item_second)
Split_Total2 <- data.frame(Total_First,Total_Second)
as.data.frame(Split_Total2, row.names = test_takers)

##            Total_First Total_Second
## Student 1            7            4
## Student 2           12            1
## Student 3           11           10
## Student 4            6            3
## Student 5           13            0
## Student 6            8            4
## Student 7           13           12
## Student 8           12           11
## Student 9           13           12
## Student 10           7            3
## Student 11          11            6
## Student 12           8            3
## Student 13           9            3
## Student 14          10            4
## Student 15          11            7
## Student 16          12            8
## Student 17           3            4
## Student 18          13            9
## Student 19           7            0
## Student 20          10           11

Correlation for the two subsets

(cor_first_second <- cor(Total_First, Total_Second))

## [1] 0.5026067

Spearman-Brown adjustment of split-half estimates

Odd-Even Split

(half <- 2 * cor_odd_even / (1 + cor_odd_even))

## [1] 0.9165757

Spearman-Brown adjustment for Odd-Even split is 0.92

First and second halves split

(half <- 2 * cor_first_second / (1 + cor_first_second))

## [1] 0.6689797

Spearman-Brown adjustment for first and second half split is 0.67

Reliability Estimates

Mubarak Mojoyinola

10/9/2021

KR20 Estimate

proportion of sample who answered item correctly (p)

proportion of sample who answered item incorrectly (q)

sum of pq

Grand sum of pq:\(\sum{pq}\)

Total score by test taker

Variance of total scores by students

KR20

Split half: Odd-Even Items

Odd items

Even items

Total score by test takers by test subset

Correlation for the two subsets

Split half first and last half Items

First half items

Second half items

Total score by test takers by test subset

Correlation for the two subsets

Spearman-Brown adjustment of split-half estimates

Odd-Even Split

First and second halves split