# install.packages("readxl") # To read Excel files
# install.packages("ggplot2") # For visual checks
# install.packages("car") Research Project Stats Analysis
Data Exploration
Install packages
Load packages
library(readxl)
library(ggplot2)Load csv data file
data <- read.csv("~/Library/Mobile Documents/com~apple~CloudDocs/Documents/MSc EPHW/MSc Research Project/Data new.csv")Structure & summaries
# View column names and types
str(data)'data.frame': 18 obs. of 27 variables:
$ Horse : chr "Banngah" "Barbie" "Bling" "Bob" ...
$ Age : num 18 17 12 18 25 20 17 18 14 11 ...
$ Sex : chr "G" "M" "M" "G" ...
$ chew.min.B : num 68 71 64 75 66 66 72 72 63 86 ...
$ X2mm.B : num 0.33 0.345 0.259 0.281 0.255 ...
$ X1mm.B : num 0.28 0.245 0.315 0.36 0.357 ...
$ X0.5mm.B : num 0.08 0.0545 0.0741 0.0674 0.0612 0.0755 0.0816 0.069 0.0976 0.0842 ...
$ X0.25mm.B : num 0.03 0.0364 0.0463 0.0225 0.0204 0.0472 0.0612 0.069 0.0732 0.0316 ...
$ X0.08mm.B : num 0.07 0.1091 0.1019 0.1011 0.0306 ...
$ longest.B : num 15.2 23 25 37.2 19.9 ...
$ QA.1 : num 11.24 14.43 8.98 11.71 6.76 ...
$ QB.1 : num 7.02 11.66 4.96 8.72 7.6 ...
$ QC.1 : num 13.9 7.75 8.57 4.75 16.01 ...
$ QD.1 : num 4.36 8.83 5.9 13.02 12.49 ...
$ shavings.present.B: chr "N" "Y" "Y" "Y" ...
$ chew.min.A : num 68 68 69 78 66 65 64 78 67 75 ...
$ X2mm.A : num 0.317 0.284 0.206 0.21 0.277 ...
$ X1mm.A : num 0.317 0.284 0.422 0.298 0.25 ...
$ X0.5mm.A : num 0.0792 0.1009 0.0784 0.0403 0.0536 ...
$ X0.25mm.A : num 0.0396 0.0459 0.0588 0.0645 0.0268 0.0283 0.0412 0.0342 0.0795 0.0254 ...
$ X0.08mm.A : num 0.0693 0.1193 0.1275 0.1855 0.0893 ...
$ longest.A : num 33.3 20.2 17.3 19.6 21.5 ...
$ QA.2 : num 11.18 13.89 11.72 5.58 13.25 ...
$ QB.2 : num 6.93 8.27 9.62 9.15 6.01 4.53 3.05 7.57 6.62 7.79 ...
$ QC.2 : num 7.44 7.44 11.59 10.49 8.77 ...
$ QD.2 : num 13.63 10.52 9.84 7.77 7.55 ...
$ shavings.present.A: chr "Y" "Y" "Y" "Y" ...# Quick look at first few rows
head(data) Horse Age Sex chew.min.B X2mm.B X1mm.B X0.5mm.B X0.25mm.B X0.08mm.B
1 Banngah 18 G 68 0.3300 0.2800 0.0800 0.0300 0.0700
2 Barbie 17 M 71 0.3455 0.2455 0.0545 0.0364 0.1091
3 Bling 12 M 64 0.2593 0.3148 0.0741 0.0463 0.1019
4 Bob 18 G 75 0.2809 0.3596 0.0674 0.0225 0.1011
5 Bobby 25 G 66 0.2551 0.3571 0.0612 0.0204 0.0306
6 Daniel 20 G 66 0.2830 0.3208 0.0755 0.0472 0.0755
longest.B QA.1 QB.1 QC.1 QD.1 shavings.present.B chew.min.A X2mm.A X1mm.A
1 15.24 11.24 7.02 13.90 4.36 N 68 0.3168 0.3168
2 22.98 14.43 11.66 7.75 8.83 Y 68 0.2844 0.2844
3 25.04 8.98 4.96 8.57 5.90 Y 69 0.2059 0.4216
4 37.17 11.71 8.72 4.75 13.02 Y 78 0.2097 0.2984
5 19.90 6.76 7.60 16.01 12.49 Y 66 0.2768 0.2500
6 22.39 5.94 9.48 7.43 22.39 Y 65 0.2358 0.3491
X0.5mm.A X0.25mm.A X0.08mm.A longest.A QA.2 QB.2 QC.2 QD.2
1 0.0792 0.0396 0.0693 33.34 11.18 6.93 7.44 13.63
2 0.1009 0.0459 0.1193 20.25 13.89 8.27 7.44 10.52
3 0.0784 0.0588 0.1275 17.31 11.72 9.62 11.59 9.84
4 0.0403 0.0645 0.1855 19.58 5.58 9.15 10.49 7.77
5 0.0536 0.0268 0.0893 21.48 13.25 6.01 8.77 7.55
6 0.1038 0.0283 0.1038 24.67 8.26 4.53 18.62 9.95
shavings.present.A
1 Y
2 Y
3 Y
4 Y
5 N
6 Y# Summary statistics
summary(data) Horse Age Sex chew.min.B
Length:18 Min. : 9.00 Length:18 Min. :62.00
Class :character 1st Qu.:14.00 Class :character 1st Qu.:64.50
Mode :character Median :17.50 Mode :character Median :70.00
Mean :16.94 Mean :70.06
3rd Qu.:20.00 3rd Qu.:72.75
Max. :25.00 Max. :86.00
X2mm.B X1mm.B X0.5mm.B X0.25mm.B
Min. :0.1917 Min. :0.2455 Min. :0.02730 Min. :0.02040
1st Qu.:0.2554 1st Qu.:0.3013 1st Qu.:0.06275 1st Qu.:0.03272
Median :0.2819 Median :0.3291 Median :0.07205 Median :0.04675
Mean :0.2827 Mean :0.3315 Mean :0.07123 Mean :0.04718
3rd Qu.:0.3206 3rd Qu.:0.3590 3rd Qu.:0.08140 3rd Qu.:0.06105
Max. :0.3636 Max. :0.4182 Max. :0.10830 Max. :0.07500
X0.08mm.B longest.B QA.1 QB.1
Min. :0.03060 Min. :15.24 Min. : 5.940 Min. : 4.960
1st Qu.:0.07333 1st Qu.:20.03 1st Qu.: 6.830 1st Qu.: 7.133
Median :0.08940 Median :22.96 Median : 8.390 Median : 7.745
Mean :0.08824 Mean :24.21 Mean : 9.714 Mean : 8.214
3rd Qu.:0.10197 3rd Qu.:27.98 3rd Qu.:11.592 3rd Qu.: 9.290
Max. :0.12730 Max. :37.17 Max. :19.920 Max. :11.690
QC.1 QD.1 shavings.present.B chew.min.A
Min. : 4.750 Min. : 3.380 Length:18 Min. :63.00
1st Qu.: 6.657 1st Qu.: 5.893 Class :character 1st Qu.:66.25
Median : 8.145 Median : 8.920 Mode :character Median :69.00
Mean : 8.943 Mean : 9.996 Mean :69.83
3rd Qu.: 9.717 3rd Qu.:12.328 3rd Qu.:72.75
Max. :16.010 Max. :22.390 Max. :78.00
X2mm.A X1mm.A X0.5mm.A X0.25mm.A
Min. :0.1705 Min. :0.2500 Min. :0.04030 Min. :0.02540
1st Qu.:0.2069 1st Qu.:0.2911 1st Qu.:0.07028 1st Qu.:0.03068
Median :0.2529 Median :0.3238 Median :0.08040 Median :0.03810
Mean :0.2543 Mean :0.3277 Mean :0.07910 Mean :0.04174
3rd Qu.:0.3041 3rd Qu.:0.3471 3rd Qu.:0.09638 3rd Qu.:0.04972
Max. :0.3390 Max. :0.4216 Max. :0.10380 Max. :0.07950
X0.08mm.A longest.A QA.2 QB.2
Min. :0.06800 Min. :16.10 Min. : 3.340 Min. : 3.050
1st Qu.:0.09248 1st Qu.:18.44 1st Qu.: 5.905 1st Qu.: 6.162
Median :0.11150 Median :21.79 Median : 8.235 Median : 7.725
Mean :0.11108 Mean :24.53 Mean : 8.740 Mean : 8.052
3rd Qu.:0.12260 3rd Qu.:27.53 3rd Qu.:11.367 3rd Qu.: 9.502
Max. :0.18550 Max. :45.05 Max. :13.890 Max. :15.000
QC.2 QD.2 shavings.present.A
Min. : 5.330 Min. : 4.680 Length:18
1st Qu.: 7.440 1st Qu.: 6.825 Class :character
Median : 8.235 Median : 7.900 Mode :character
Mean : 9.595 Mean : 8.404
3rd Qu.:11.315 3rd Qu.: 9.950
Max. :18.670 Max. :13.630 Missing values
# Count missing values per column
colSums(is.na(data)) Horse Age Sex chew.min.B
0 0 0 0
X2mm.B X1mm.B X0.5mm.B X0.25mm.B
0 0 0 0
X0.08mm.B longest.B QA.1 QB.1
0 0 0 0
QC.1 QD.1 shavings.present.B chew.min.A
0 0 0 0
X2mm.A X1mm.A X0.5mm.A X0.25mm.A
0 0 0 0
X0.08mm.A longest.A QA.2 QB.2
0 0 0 0
QC.2 QD.2 shavings.present.A
0 0 0 # Visualize missing data
# install.packages("naniar")
library(naniar)
vis_miss(data)
sapply(data, class) Horse Age Sex chew.min.B
"character" "numeric" "character" "numeric"
X2mm.B X1mm.B X0.5mm.B X0.25mm.B
"numeric" "numeric" "numeric" "numeric"
X0.08mm.B longest.B QA.1 QB.1
"numeric" "numeric" "numeric" "numeric"
QC.1 QD.1 shavings.present.B chew.min.A
"numeric" "numeric" "character" "numeric"
X2mm.A X1mm.A X0.5mm.A X0.25mm.A
"numeric" "numeric" "numeric" "numeric"
X0.08mm.A longest.A QA.2 QB.2
"numeric" "numeric" "numeric" "numeric"
QC.2 QD.2 shavings.present.A
"numeric" "numeric" "character" # Convert character to factor if needed
data$Sex <- as.factor(data$Sex)
data$Horse <- as.factor(data$Horse)
data$shavings.present.B <- as.factor(data$shavings.present.B)
data$shavings.present.A <- as.factor(data$shavings.present.A)Outliers
# Age
boxplot(data$Age, main = "Age outliers")
boxplot(data$chew.min.B, main = "Chew/min B outliers")
boxplot(data$chew.min.A, main = "Chew/min A outliers")
boxplot(data$X1mm.B, main = "1mm B outliers")
Normality check
shapiro.test(data$Age)
Shapiro-Wilk normality test
data: data$Age
W = 0.98126, p-value = 0.962shapiro.test(data$chew.min.B)
Shapiro-Wilk normality test
data: data$chew.min.B
W = 0.92399, p-value = 0.1521shapiro.test(data$X2mm.B)
Shapiro-Wilk normality test
data: data$X2mm.B
W = 0.96569, p-value = 0.714shapiro.test(data$X1mm.B)
Shapiro-Wilk normality test
data: data$X1mm.B
W = 0.97987, p-value = 0.9487shapiro.test(data$X0.5mm.B)
Shapiro-Wilk normality test
data: data$X0.5mm.B
W = 0.96658, p-value = 0.7314shapiro.test(data$X0.25mm.B)
Shapiro-Wilk normality test
data: data$X0.25mm.B
W = 0.94309, p-value = 0.327shapiro.test(data$X0.08mm.B)
Shapiro-Wilk normality test
data: data$X0.08mm.B
W = 0.9478, p-value = 0.3916shapiro.test(data$longest.B)
Shapiro-Wilk normality test
data: data$longest.B
W = 0.93712, p-value = 0.2586shapiro.test(data$QA.1) # Not normal
Shapiro-Wilk normality test
data: data$QA.1
W = 0.83151, p-value = 0.004387shapiro.test(data$QB.1)
Shapiro-Wilk normality test
data: data$QB.1
W = 0.94606, p-value = 0.3666shapiro.test(data$QC.1) # Not normal
Shapiro-Wilk normality test
data: data$QC.1
W = 0.89236, p-value = 0.04237shapiro.test(data$QD.1)
Shapiro-Wilk normality test
data: data$QD.1
W = 0.91078, p-value = 0.08879shapiro.test(data$chew.min.A)
Shapiro-Wilk normality test
data: data$chew.min.A
W = 0.94147, p-value = 0.307shapiro.test(data$X2mm.A)
Shapiro-Wilk normality test
data: data$X2mm.A
W = 0.93801, p-value = 0.2678shapiro.test(data$X1mm.A)
Shapiro-Wilk normality test
data: data$X1mm.A
W = 0.95726, p-value = 0.5497shapiro.test(data$X0.5mm.A)
Shapiro-Wilk normality test
data: data$X0.5mm.A
W = 0.91568, p-value = 0.1084shapiro.test(data$X0.25mm.A) # Not normal BUT rounded up is 0.05?
Shapiro-Wilk normality test
data: data$X0.25mm.A
W = 0.89482, p-value = 0.04671shapiro.test(data$X0.08mm.A)
Shapiro-Wilk normality test
data: data$X0.08mm.A
W = 0.94548, p-value = 0.3586shapiro.test(data$longest.A) # Not normal
Shapiro-Wilk normality test
data: data$longest.A
W = 0.85861, p-value = 0.01161shapiro.test(data$QA.2)
Shapiro-Wilk normality test
data: data$QA.2
W = 0.9547, p-value = 0.5034shapiro.test(data$QB.2)
Shapiro-Wilk normality test
data: data$QB.2
W = 0.9655, p-value = 0.7102shapiro.test(data$QC.2) # Not normal
Shapiro-Wilk normality test
data: data$QC.2
W = 0.83384, p-value = 0.004759shapiro.test(data$QD.2)
Shapiro-Wilk normality test
data: data$QD.2
W = 0.95269, p-value = 0.4687Dealing with non-normal data
| Goal | Parametric Test | Non-Parametric Alternative |
|---|---|---|
| Compare 2 independent groups | t.test() |
wilcox.test() (Mann–Whitney U test) |
| Compare 2 related groups | paired t-test |
wilcox.test(paired = TRUE) |
| Compare >2 groups (independent) | anova() |
kruskal.test() |
| Compare >2 related groups | repeated measures ANOVA |
Friedman test (friedman.test()) |
| Correlation | cor(method = "pearson") |
cor(method = "spearman") |
Older horse age factor (>17)
table(data$Age)
9 11 12 14 15 17 18 20 21 22 25
1 1 1 3 1 2 3 3 1 1 1 older_horses <- subset(data, Age > 17)
library(dplyr)
Attaching package: 'dplyr'The following objects are masked from 'package:stats':
filter, lagThe following objects are masked from 'package:base':
intersect, setdiff, setequal, unionolder_horses <- data %>% filter(Age > 17)
View(older_horses)
summary(older_horses) Horse Age Sex chew.min.B X2mm.B
Banngah:1 Min. :18.00 G:8 Min. :63.00 Min. :0.2119
Bob :1 1st Qu.:18.00 M:1 1st Qu.:66.00 1st Qu.:0.2551
Bobby :1 Median :20.00 Median :70.00 Median :0.2830
Daniel :1 Mean :20.22 Mean :69.89 Mean :0.2802
Greta :1 3rd Qu.:21.00 3rd Qu.:73.00 3rd Qu.:0.3146
Misty :1 Max. :25.00 Max. :76.00 Max. :0.3300
(Other):3
X1mm.B X0.5mm.B X0.25mm.B X0.08mm.B
Min. :0.2800 Min. :0.0333 Min. :0.02040 Min. :0.03060
1st Qu.:0.3208 1st Qu.:0.0674 1st Qu.:0.02250 1st Qu.:0.07000
Median :0.3305 Median :0.0690 Median :0.04440 Median :0.08890
Mean :0.3347 Mean :0.0695 Mean :0.04059 Mean :0.08126
3rd Qu.:0.3571 3rd Qu.:0.0755 3rd Qu.:0.05000 3rd Qu.:0.09320
Max. :0.4000 Max. :0.1017 Max. :0.06900 Max. :0.11210
longest.B QA.1 QB.1 QC.1
Min. :15.24 Min. : 5.940 Min. :7.020 Min. : 4.750
1st Qu.:16.28 1st Qu.: 7.040 1st Qu.:7.600 1st Qu.: 6.400
Median :20.42 Median : 8.680 Median :7.780 Median : 7.620
Mean :22.90 Mean : 9.768 Mean :8.161 Mean : 9.427
3rd Qu.:25.10 3rd Qu.:11.710 3rd Qu.:8.720 3rd Qu.:13.900
Max. :37.17 Max. :16.970 Max. :9.700 Max. :16.010
QD.1 shavings.present.B chew.min.A X2mm.A
Min. : 3.380 N:2 Min. :63.00 Min. :0.1837
1st Qu.: 5.400 Y:7 1st Qu.:66.00 1st Qu.:0.2358
Median :11.250 Median :69.00 Median :0.2578
Mean : 9.949 Mean :70.56 Mean :0.2629
3rd Qu.:12.490 3rd Qu.:76.00 3rd Qu.:0.3107
Max. :22.390 Max. :78.00 Max. :0.3265
X1mm.A X0.5mm.A X0.25mm.A X0.08mm.A
Min. :0.2500 Min. :0.04030 Min. :0.02680 Min. :0.0680
1st Qu.:0.2984 1st Qu.:0.07770 1st Qu.:0.03060 1st Qu.:0.0893
Median :0.3203 Median :0.07920 Median :0.03880 Median :0.1038
Mean :0.3266 Mean :0.07798 Mean :0.04094 Mean :0.1119
3rd Qu.:0.3491 3rd Qu.:0.08550 3rd Qu.:0.05100 3rd Qu.:0.1368
Max. :0.4078 Max. :0.10380 Max. :0.06450 Max. :0.1855
longest.A QA.2 QB.2 QC.2
Min. :19.58 Min. : 5.580 Min. :4.530 Min. : 5.52
1st Qu.:21.48 1st Qu.: 6.850 1st Qu.:5.890 1st Qu.: 7.65
Median :24.67 Median : 9.490 Median :6.930 Median : 8.89
Mean :27.27 Mean : 9.273 Mean :7.096 Mean :10.20
3rd Qu.:31.55 3rd Qu.:11.180 3rd Qu.:8.570 3rd Qu.:12.06
Max. :45.05 Max. :13.250 Max. :9.980 Max. :18.62
QD.2 shavings.present.A
Min. : 7.260 N:3
1st Qu.: 7.550 Y:6
Median : 9.950
Mean : 9.638
3rd Qu.:10.480
Max. :13.630
shapiro.test(older_horses$Age)
Shapiro-Wilk normality test
data: older_horses$Age
W = 0.86985, p-value = 0.1225shapiro.test(older_horses$chew.min.B)
Shapiro-Wilk normality test
data: older_horses$chew.min.B
W = 0.95819, p-value = 0.7792shapiro.test(older_horses$X2mm.B)
Shapiro-Wilk normality test
data: older_horses$X2mm.B
W = 0.94872, p-value = 0.6761shapiro.test(older_horses$X1mm.B)
Shapiro-Wilk normality test
data: older_horses$X1mm.B
W = 0.97639, p-value = 0.9432shapiro.test(older_horses$X0.5mm.B)
Shapiro-Wilk normality test
data: older_horses$X0.5mm.B
W = 0.90388, p-value = 0.2754shapiro.test(older_horses$X0.25mm.B)
Shapiro-Wilk normality test
data: older_horses$X0.25mm.B
W = 0.91534, p-value = 0.3551shapiro.test(older_horses$X0.08mm.B)
Shapiro-Wilk normality test
data: older_horses$X0.08mm.B
W = 0.91642, p-value = 0.3635shapiro.test(older_horses$longest.B)
Shapiro-Wilk normality test
data: older_horses$longest.B
W = 0.86939, p-value = 0.1211shapiro.test(older_horses$QA.1) # Not normal
Shapiro-Wilk normality test
data: older_horses$QA.1
W = 0.89649, p-value = 0.2324shapiro.test(older_horses$QB.1)
Shapiro-Wilk normality test
data: older_horses$QB.1
W = 0.91649, p-value = 0.364shapiro.test(older_horses$QC.1)
Shapiro-Wilk normality test
data: older_horses$QC.1
W = 0.85315, p-value = 0.08071shapiro.test(older_horses$QD.1) # Not normal
Shapiro-Wilk normality test
data: older_horses$QD.1
W = 0.87954, p-value = 0.1553shapiro.test(older_horses$chew.min.A)
Shapiro-Wilk normality test
data: older_horses$chew.min.A
W = 0.91229, p-value = 0.3322shapiro.test(older_horses$X2mm.A)
Shapiro-Wilk normality test
data: older_horses$X2mm.A
W = 0.95306, p-value = 0.7236shapiro.test(older_horses$X1mm.A)
Shapiro-Wilk normality test
data: older_horses$X1mm.A
W = 0.99023, p-value = 0.9964shapiro.test(older_horses$X0.5mm.A)
Shapiro-Wilk normality test
data: older_horses$X0.5mm.A
W = 0.90716, p-value = 0.2965shapiro.test(older_horses$X0.25mm.A)
Shapiro-Wilk normality test
data: older_horses$X0.25mm.A
W = 0.91403, p-value = 0.3451shapiro.test(older_horses$X0.08mm.A)
Shapiro-Wilk normality test
data: older_horses$X0.08mm.A
W = 0.93168, p-value = 0.4975shapiro.test(older_horses$longest.A) # Not normal
Shapiro-Wilk normality test
data: older_horses$longest.A
W = 0.85734, p-value = 0.08969shapiro.test(older_horses$QA.2)
Shapiro-Wilk normality test
data: older_horses$QA.2
W = 0.9309, p-value = 0.49shapiro.test(older_horses$QB.2)
Shapiro-Wilk normality test
data: older_horses$QB.2
W = 0.96213, p-value = 0.8203shapiro.test(older_horses$QC.2) # Not normal
Shapiro-Wilk normality test
data: older_horses$QC.2
W = 0.89959, p-value = 0.2496shapiro.test(older_horses$QD.2)
Shapiro-Wilk normality test
data: older_horses$QD.2
W = 0.8701, p-value = 0.1233Younger horses age factor (<= 17)
younger_horses <- subset(data, Age <= 17)
library(dplyr)
younger_horses <- data %>% filter(Age <= 17)
View(younger_horses)
summary(older_horses) Horse Age Sex chew.min.B X2mm.B
Banngah:1 Min. :18.00 G:8 Min. :63.00 Min. :0.2119
Bob :1 1st Qu.:18.00 M:1 1st Qu.:66.00 1st Qu.:0.2551
Bobby :1 Median :20.00 Median :70.00 Median :0.2830
Daniel :1 Mean :20.22 Mean :69.89 Mean :0.2802
Greta :1 3rd Qu.:21.00 3rd Qu.:73.00 3rd Qu.:0.3146
Misty :1 Max. :25.00 Max. :76.00 Max. :0.3300
(Other):3
X1mm.B X0.5mm.B X0.25mm.B X0.08mm.B
Min. :0.2800 Min. :0.0333 Min. :0.02040 Min. :0.03060
1st Qu.:0.3208 1st Qu.:0.0674 1st Qu.:0.02250 1st Qu.:0.07000
Median :0.3305 Median :0.0690 Median :0.04440 Median :0.08890
Mean :0.3347 Mean :0.0695 Mean :0.04059 Mean :0.08126
3rd Qu.:0.3571 3rd Qu.:0.0755 3rd Qu.:0.05000 3rd Qu.:0.09320
Max. :0.4000 Max. :0.1017 Max. :0.06900 Max. :0.11210
longest.B QA.1 QB.1 QC.1
Min. :15.24 Min. : 5.940 Min. :7.020 Min. : 4.750
1st Qu.:16.28 1st Qu.: 7.040 1st Qu.:7.600 1st Qu.: 6.400
Median :20.42 Median : 8.680 Median :7.780 Median : 7.620
Mean :22.90 Mean : 9.768 Mean :8.161 Mean : 9.427
3rd Qu.:25.10 3rd Qu.:11.710 3rd Qu.:8.720 3rd Qu.:13.900
Max. :37.17 Max. :16.970 Max. :9.700 Max. :16.010
QD.1 shavings.present.B chew.min.A X2mm.A
Min. : 3.380 N:2 Min. :63.00 Min. :0.1837
1st Qu.: 5.400 Y:7 1st Qu.:66.00 1st Qu.:0.2358
Median :11.250 Median :69.00 Median :0.2578
Mean : 9.949 Mean :70.56 Mean :0.2629
3rd Qu.:12.490 3rd Qu.:76.00 3rd Qu.:0.3107
Max. :22.390 Max. :78.00 Max. :0.3265
X1mm.A X0.5mm.A X0.25mm.A X0.08mm.A
Min. :0.2500 Min. :0.04030 Min. :0.02680 Min. :0.0680
1st Qu.:0.2984 1st Qu.:0.07770 1st Qu.:0.03060 1st Qu.:0.0893
Median :0.3203 Median :0.07920 Median :0.03880 Median :0.1038
Mean :0.3266 Mean :0.07798 Mean :0.04094 Mean :0.1119
3rd Qu.:0.3491 3rd Qu.:0.08550 3rd Qu.:0.05100 3rd Qu.:0.1368
Max. :0.4078 Max. :0.10380 Max. :0.06450 Max. :0.1855
longest.A QA.2 QB.2 QC.2
Min. :19.58 Min. : 5.580 Min. :4.530 Min. : 5.52
1st Qu.:21.48 1st Qu.: 6.850 1st Qu.:5.890 1st Qu.: 7.65
Median :24.67 Median : 9.490 Median :6.930 Median : 8.89
Mean :27.27 Mean : 9.273 Mean :7.096 Mean :10.20
3rd Qu.:31.55 3rd Qu.:11.180 3rd Qu.:8.570 3rd Qu.:12.06
Max. :45.05 Max. :13.250 Max. :9.980 Max. :18.62
QD.2 shavings.present.A
Min. : 7.260 N:3
1st Qu.: 7.550 Y:6
Median : 9.950
Mean : 9.638
3rd Qu.:10.480
Max. :13.630
shapiro.test(younger_horses$Age)
Shapiro-Wilk normality test
data: younger_horses$Age
W = 0.93849, p-value = 0.5661shapiro.test(younger_horses$chew.min.B)
Shapiro-Wilk normality test
data: younger_horses$chew.min.B
W = 0.86687, p-value = 0.1138shapiro.test(younger_horses$X2mm.B)
Shapiro-Wilk normality test
data: younger_horses$X2mm.B
W = 0.93025, p-value = 0.4837shapiro.test(younger_horses$X1mm.B)
Shapiro-Wilk normality test
data: younger_horses$X1mm.B
W = 0.95345, p-value = 0.7279shapiro.test(younger_horses$X0.5mm.B)
Shapiro-Wilk normality test
data: younger_horses$X0.5mm.B
W = 0.95806, p-value = 0.7778shapiro.test(younger_horses$X0.25mm.B)
Shapiro-Wilk normality test
data: younger_horses$X0.25mm.B
W = 0.90043, p-value = 0.2545shapiro.test(younger_horses$X0.08mm.B)
Shapiro-Wilk normality test
data: younger_horses$X0.08mm.B
W = 0.89198, p-value = 0.2091shapiro.test(younger_horses$longest.B)
Shapiro-Wilk normality test
data: younger_horses$longest.B
W = 0.9899, p-value = 0.996shapiro.test(younger_horses$QA.1) # Not normal BUT rounded up = 0.05
Shapiro-Wilk normality test
data: younger_horses$QA.1
W = 0.74688, p-value = 0.004988shapiro.test(younger_horses$QB.1)
Shapiro-Wilk normality test
data: younger_horses$QB.1
W = 0.89386, p-value = 0.2185shapiro.test(younger_horses$QC.1)
Shapiro-Wilk normality test
data: younger_horses$QC.1
W = 0.94953, p-value = 0.6849shapiro.test(younger_horses$QD.1) # Not normal
Shapiro-Wilk normality test
data: younger_horses$QD.1
W = 0.81495, p-value = 0.03021shapiro.test(younger_horses$chew.min.A)
Shapiro-Wilk normality test
data: younger_horses$chew.min.A
W = 0.97794, p-value = 0.9529shapiro.test(younger_horses$X2mm.A)
Shapiro-Wilk normality test
data: younger_horses$X2mm.A
W = 0.92091, p-value = 0.3998shapiro.test(younger_horses$X1mm.A)
Shapiro-Wilk normality test
data: younger_horses$X1mm.A
W = 0.88376, p-value = 0.1719shapiro.test(younger_horses$X0.5mm.A)
Shapiro-Wilk normality test
data: younger_horses$X0.5mm.A
W = 0.90584, p-value = 0.2878shapiro.test(younger_horses$X0.25mm.A)
Shapiro-Wilk normality test
data: younger_horses$X0.25mm.A
W = 0.87347, p-value = 0.1339shapiro.test(younger_horses$X0.08mm.A)
Shapiro-Wilk normality test
data: younger_horses$X0.08mm.A
W = 0.83943, p-value = 0.05691shapiro.test(younger_horses$longest.A) # Not normal
Shapiro-Wilk normality test
data: younger_horses$longest.A
W = 0.73884, p-value = 0.004023shapiro.test(younger_horses$QA.2)
Shapiro-Wilk normality test
data: younger_horses$QA.2
W = 0.95227, p-value = 0.715shapiro.test(younger_horses$QB.2)
Shapiro-Wilk normality test
data: younger_horses$QB.2
W = 0.97127, p-value = 0.9054shapiro.test(younger_horses$QC.2) # Not normal
Shapiro-Wilk normality test
data: younger_horses$QC.2
W = 0.7771, p-value = 0.01115shapiro.test(younger_horses$QD.2)
Shapiro-Wilk normality test
data: younger_horses$QD.2
W = 0.90644, p-value = 0.2917Adding ‘younger’ or ‘older’ groups to the data
data$group <- ifelse(data$Age > 17, "older", "younger")
table(data$group)
older younger
9 9 View(data)What do we want to test:
Compare each sieve DM proportion against one another (before and after) = paired t-test
Compare each sieve DM proportion against younger and older horses = t-test
Compare each sieve DM proportion within an age group before and after? = paired t-test
Compare each quadrant particle length before and after = repeated measures ANOVA/ Friedman (due to non-normal parameters)
Compare each quadrant particle length before and after between older and younger = ANOVA/Kruskal test (due to non-normal parameters)
Compare each quadrant particle length before and after between within the age groups = repeated measures ANOVA/Friedman test (due to non-normal parameters)
Compare longest particle before and after = paired t-test
Compare shavings before and after = paired t-test
Compare chew rates (chew/min) before and after = paired t-test
Compare chew rates between age groups = t-test
Compare chew rates within age groups before and after = paired t-test
1) Comparing each sieves DM proportions before & after dental treatment (all horses)
t.test(data$X2mm.B, data$X2mm.A, paired = TRUE) # Signif diff (increase in proportion of particles on sieve after dental)
Paired t-test
data: data$X2mm.B and data$X2mm.A
t = 2.5873, df = 17, p-value = 0.01918
alternative hypothesis: true mean difference is not equal to 0
95 percent confidence interval:
0.00523437 0.05148785
sample estimates:
mean difference
0.02836111 t.test(data$X1mm.B, data$X1mm.A, paired = TRUE) # Not signif
Paired t-test
data: data$X1mm.B and data$X1mm.A
t = 0.31063, df = 17, p-value = 0.7599
alternative hypothesis: true mean difference is not equal to 0
95 percent confidence interval:
-0.02204213 0.02965325
sample estimates:
mean difference
0.003805556 t.test(data$X0.5mm.B, data$X0.5mm.A, paired = TRUE) # Not signif
Paired t-test
data: data$X0.5mm.B and data$X0.5mm.A
t = -1.2542, df = 17, p-value = 0.2267
alternative hypothesis: true mean difference is not equal to 0
95 percent confidence interval:
-0.021114828 0.005370383
sample estimates:
mean difference
-0.007872222 t.test(data$X0.25mm.B, data$X0.25mm.A, paired = TRUE) # Not signif
Paired t-test
data: data$X0.25mm.B and data$X0.25mm.A
t = 1.1096, df = 17, p-value = 0.2826
alternative hypothesis: true mean difference is not equal to 0
95 percent confidence interval:
-0.004907393 0.015796282
sample estimates:
mean difference
0.005444444 t.test(data$X0.08mm.B, data$X0.08mm.A, paired = TRUE) # Signif diff (decrease in proportion of particles on sieve after dental)
Paired t-test
data: data$X0.08mm.B and data$X0.08mm.A
t = -3.6307, df = 17, p-value = 0.002067
alternative hypothesis: true mean difference is not equal to 0
95 percent confidence interval:
-0.036110729 -0.009567048
sample estimates:
mean difference
-0.02283889 2) Comparing each sieves DM proportions between age groups
Before dentist:
t.test(X2mm.B ~ group, data = data) # Not signif
Welch Two Sample t-test
data: X2mm.B by group
t = -0.19833, df = 13.508, p-value = 0.8457
alternative hypothesis: true difference in means between group older and group younger is not equal to 0
95 percent confidence interval:
-0.05767513 0.04794180
sample estimates:
mean in group older mean in group younger
0.2802444 0.2851111 t.test(X1mm.B ~ group, data = data) # Not signif
Welch Two Sample t-test
data: X1mm.B by group
t = 0.28775, df = 13.112, p-value = 0.778
alternative hypothesis: true difference in means between group older and group younger is not equal to 0
95 percent confidence interval:
-0.04247502 0.05554168
sample estimates:
mean in group older mean in group younger
0.3347444 0.3282111 t.test(X0.5mm.B ~ group, data = data) # Not signif
Welch Two Sample t-test
data: X0.5mm.B by group
t = -0.33349, df = 14.389, p-value = 0.7436
alternative hypothesis: true difference in means between group older and group younger is not equal to 0
95 percent confidence interval:
-0.02562289 0.01871178
sample estimates:
mean in group older mean in group younger
0.06950000 0.07295556 t.test(X0.25mm.B ~ group, data = data) # Not signif
Welch Two Sample t-test
data: X0.25mm.B by group
t = -1.6383, df = 15.938, p-value = 0.1209
alternative hypothesis: true difference in means between group older and group younger is not equal to 0
95 percent confidence interval:
-0.030260271 0.003882493
sample estimates:
mean in group older mean in group younger
0.04058889 0.05377778 t.test(X0.08mm.B ~ group, data = data) # Not signif
Welch Two Sample t-test
data: X0.08mm.B by group
t = -1.331, df = 15.74, p-value = 0.2022
alternative hypothesis: true difference in means between group older and group younger is not equal to 0
95 percent confidence interval:
-0.036242245 0.008308912
sample estimates:
mean in group older mean in group younger
0.08125556 0.09522222 After dentist:
t.test(X2mm.A ~ group, data = data) # Not signif
Welch Two Sample t-test
data: X2mm.A by group
t = 0.64783, df = 15.262, p-value = 0.5267
alternative hypothesis: true difference in means between group older and group younger is not equal to 0
95 percent confidence interval:
-0.03902693 0.07318248
sample estimates:
mean in group older mean in group younger
0.2628556 0.2457778 t.test(X1mm.A ~ group, data = data) # Not signif
Welch Two Sample t-test
data: X1mm.A by group
t = -0.10117, df = 15.976, p-value = 0.9207
alternative hypothesis: true difference in means between group older and group younger is not equal to 0
95 percent confidence interval:
-0.04903603 0.04456936
sample estimates:
mean in group older mean in group younger
0.3265556 0.3287889 t.test(X0.5mm.A ~ group, data = data) # Not signif
Welch Two Sample t-test
data: X0.5mm.A by group
t = -0.23995, df = 15.948, p-value = 0.8134
alternative hypothesis: true difference in means between group older and group younger is not equal to 0
95 percent confidence interval:
-0.02207901 0.01759012
sample estimates:
mean in group older mean in group younger
0.07797778 0.08022222 t.test(X0.25mm.A ~ group, data = data) # Not signif
Welch Two Sample t-test
data: X0.25mm.A by group
t = -0.22066, df = 14.919, p-value = 0.8284
alternative hypothesis: true difference in means between group older and group younger is not equal to 0
95 percent confidence interval:
-0.01694411 0.01376633
sample estimates:
mean in group older mean in group younger
0.04094444 0.04253333 t.test(X0.08mm.A ~ group, data = data) # Not signif
Welch Two Sample t-test
data: X0.08mm.A by group
t = 0.11844, df = 10.828, p-value = 0.9079
alternative hypothesis: true difference in means between group older and group younger is not equal to 0
95 percent confidence interval:
-0.02858166 0.03182610
sample estimates:
mean in group older mean in group younger
0.1118889 0.1102667 3) Comparing each sieves DM proportion within an age group before & after dental treatment
Older horses:
t.test(older_horses$X2mm.B, older_horses$X2mm.A, paired = TRUE) # Not signif
Paired t-test
data: older_horses$X2mm.B and older_horses$X2mm.A
t = 1.1437, df = 8, p-value = 0.2858
alternative hypothesis: true mean difference is not equal to 0
95 percent confidence interval:
-0.01767125 0.05244903
sample estimates:
mean difference
0.01738889 t.test(older_horses$X1mm.B, older_horses$X1mm.A, paired = TRUE) # Not signif
Paired t-test
data: older_horses$X1mm.B and older_horses$X1mm.A
t = 0.46706, df = 8, p-value = 0.6529
alternative hypothesis: true mean difference is not equal to 0
95 percent confidence interval:
-0.03224229 0.04862007
sample estimates:
mean difference
0.008188889 t.test(older_horses$X0.5mm.B, older_horses$X0.5mm.A, paired = TRUE) # Not signif
Paired t-test
data: older_horses$X0.5mm.B and older_horses$X0.5mm.A
t = -1.2182, df = 8, p-value = 0.2579
alternative hypothesis: true mean difference is not equal to 0
95 percent confidence interval:
-0.024526095 0.007570539
sample estimates:
mean difference
-0.008477778 t.test(older_horses$X0.25mm.B, older_horses$X0.25mm.A, paired = TRUE) # Not signif
Paired t-test
data: older_horses$X0.25mm.B and older_horses$X0.25mm.A
t = -0.048748, df = 8, p-value = 0.9623
alternative hypothesis: true mean difference is not equal to 0
95 percent confidence interval:
-0.01717486 0.01646375
sample estimates:
mean difference
-0.0003555556 t.test(older_horses$X0.08mm.B, older_horses$X0.08mm.A, paired = TRUE) # Signif diff (decrease in proportion of particles on sieve after dental treatment)
Paired t-test
data: older_horses$X0.08mm.B and older_horses$X0.08mm.A
t = -3.2055, df = 8, p-value = 0.01251
alternative hypothesis: true mean difference is not equal to 0
95 percent confidence interval:
-0.052670893 -0.008595773
sample estimates:
mean difference
-0.03063333 Younger horses:
t.test(younger_horses$X2mm.B, younger_horses$X2mm.A, paired = TRUE) # Signif diff (increase in proportion of particle on sieve after dental treatment)
Paired t-test
data: younger_horses$X2mm.B and younger_horses$X2mm.A
t = 2.4906, df = 8, p-value = 0.03748
alternative hypothesis: true mean difference is not equal to 0
95 percent confidence interval:
0.002915844 0.075750822
sample estimates:
mean difference
0.03933333 t.test(younger_horses$X1mm.B, younger_horses$X1mm.A, paired = TRUE) # Not signif
Paired t-test
data: younger_horses$X1mm.B and younger_horses$X1mm.A
t = -0.032016, df = 8, p-value = 0.9752
alternative hypothesis: true mean difference is not equal to 0
95 percent confidence interval:
-0.04219313 0.04103757
sample estimates:
mean difference
-0.0005777778 t.test(younger_horses$X0.5mm.B, younger_horses$X0.5mm.A, paired = TRUE) # Not signif
Paired t-test
data: younger_horses$X0.5mm.B and younger_horses$X0.5mm.A
t = -0.66638, df = 8, p-value = 0.5239
alternative hypothesis: true mean difference is not equal to 0
95 percent confidence interval:
-0.03241278 0.01787944
sample estimates:
mean difference
-0.007266667 t.test(younger_horses$X0.25mm.B, younger_horses$X0.25mm.A, paired = TRUE) # Not signif
Paired t-test
data: younger_horses$X0.25mm.B and younger_horses$X0.25mm.A
t = 1.7624, df = 8, p-value = 0.116
alternative hypothesis: true mean difference is not equal to 0
95 percent confidence interval:
-0.003468127 0.025957015
sample estimates:
mean difference
0.01124444 t.test(younger_horses$X0.08mm.B, younger_horses$X0.08mm.A, paired = TRUE) # Not signif
Paired t-test
data: younger_horses$X0.08mm.B and younger_horses$X0.08mm.A
t = -1.9159, df = 8, p-value = 0.09169
alternative hypothesis: true mean difference is not equal to 0
95 percent confidence interval:
-0.033151844 0.003062955
sample estimates:
mean difference
-0.01504444 4) Comparing each quadrants particles before and after dentist (all horses)
Before dentist (1):
friedman.test(y = as.matrix(data[, c("QA.1", "QB.1", "QC.1", "QD.1")])) # No signif difference between particle lengths before
Friedman rank sum test
data: as.matrix(data[, c("QA.1", "QB.1", "QC.1", "QD.1")])
Friedman chi-squared = 0.92179, df = 3, p-value = 0.8202After dentist (2):
friedman.test(y = as.matrix(data[, c("QA.2", "QB.2", "QC.2", "QD.2")])) # No signif difference between particle lengths after dentist
Friedman rank sum test
data: as.matrix(data[, c("QA.2", "QB.2", "QC.2", "QD.2")])
Friedman chi-squared = 2.8667, df = 3, p-value = 0.4126library(dplyr)
library(tidyr)
# Create long-format dataframe with quadrants and time labels
quad_data <- data %>%
select(QA.1, QB.1, QC.1, QD.1, QA.2, QB.2, QC.2, QD.2) %>%
mutate(HorseID = 1:n()) %>%
pivot_longer(
cols = -HorseID,
names_to = "Quadrant_Time",
values_to = "Value"
) %>%
mutate(
Time = ifelse(grepl("\\.1$", Quadrant_Time), "Before", "After"),
Quadrant = substr(Quadrant_Time, 1, 2)
)
kruskal.test(Value ~ Time, data = quad_data) # No signif difference in the median particle lengths across all quadrants before and after treatment.
Kruskal-Wallis rank sum test
data: Value by Time
Kruskal-Wallis chi-squared = 0.2698, df = 1, p-value = 0.60355) Comparing each quadrants particle length before and after dental treatment between older and younger horses
data$QA.diff <- data$QA.2 - data$QA.1
kruskal.test(QA.diff ~ group, data = data) # Not signif
Kruskal-Wallis rank sum test
data: QA.diff by group
Kruskal-Wallis chi-squared = 0.23587, df = 1, p-value = 0.6272data$QB.diff <- data$QB.2 - data$QB.1
kruskal.test(QB.diff ~ group, data = data) # Not signif
Kruskal-Wallis rank sum test
data: QB.diff by group
Kruskal-Wallis chi-squared = 0.56335, df = 1, p-value = 0.4529data$QC.diff <- data$QC.2 - data$QC.1
kruskal.test(QC.diff ~ group, data = data) # Not signif
Kruskal-Wallis rank sum test
data: QC.diff by group
Kruskal-Wallis chi-squared = 0.0019493, df = 1, p-value = 0.9648data$QD.diff <- data$QD.2 - data$QD.1
kruskal.test(QD.diff ~ group, data = data) # Not signif
Kruskal-Wallis rank sum test
data: QD.diff by group
Kruskal-Wallis chi-squared = 0.4386, df = 1, p-value = 0.5078library(dplyr)
library(tidyr)
data$HorseID <- 1:nrow(data)
quad_data <- data %>%
select(HorseID, group, QA.1, QB.1, QC.1, QD.1, QA.2, QB.2, QC.2, QD.2) %>%
pivot_longer(
cols = starts_with("Q"),
names_to = "Quadrant_Time",
values_to = "Value"
) %>%
mutate(
Time = ifelse(grepl("\\.1$", Quadrant_Time), "Before", "After"),
Quadrant = substr(Quadrant_Time, 1, 2)
)
kruskal.test(Value ~ group, data = quad_data, subset = Time == "Before") # No signif diff between quadrants before dentist & age groups
Kruskal-Wallis rank sum test
data: Value by group
Kruskal-Wallis chi-squared = 0.016775, df = 1, p-value = 0.8969kruskal.test(Value ~ group, data = quad_data, subset = Time == "After") # # No signif diff between quadrants after & age groups
Kruskal-Wallis rank sum test
data: Value by group
Kruskal-Wallis chi-squared = 1.3588, df = 1, p-value = 0.24376) Comparing each quadrants particle length before and after within age groups
Older horses:
# Before:
friedman.test(as.matrix(older_horses[, c("QA.1", "QB.1", "QC.1", "QD.1")])) # There is no significant difference in median particle length across the four quadrants (QA–QD) before the dental treatment in older horses.
Friedman rank sum test
data: as.matrix(older_horses[, c("QA.1", "QB.1", "QC.1", "QD.1")])
Friedman chi-squared = 0.91011, df = 3, p-value = 0.823# After:
friedman.test(as.matrix(older_horses[, c("QA.2", "QB.2", "QC.2", "QD.2")])) # No signif diff in median particle length
Friedman rank sum test
data: as.matrix(older_horses[, c("QA.2", "QB.2", "QC.2", "QD.2")])
Friedman chi-squared = 4.0667, df = 3, p-value = 0.2544Younger horses:
# Before:
friedman.test(as.matrix(younger_horses[, c("QA.1", "QB.1", "QC.1", "QD.1")])) # No signif diff in median particle length
Friedman rank sum test
data: as.matrix(younger_horses[, c("QA.1", "QB.1", "QC.1", "QD.1")])
Friedman chi-squared = 0.86667, df = 3, p-value = 0.8335# After:
friedman.test(as.matrix(younger_horses[, c("QA.2", "QB.2", "QC.2", "QD.2")])) # No signif diff in median particle legth
Friedman rank sum test
data: as.matrix(younger_horses[, c("QA.2", "QB.2", "QC.2", "QD.2")])
Friedman chi-squared = 1.6667, df = 3, p-value = 0.6444###7) Comparing longest particle before and after dentist
After is not normally distributed thus,:
wilcox.test(data$longest.B, data$longest.A, paired = TRUE) # No significant difference in the median longest particle length before vs after dental treatment.
Wilcoxon signed rank exact test
data: data$longest.B and data$longest.A
V = 82, p-value = 0.8986
alternative hypothesis: true location shift is not equal to 08) Compare shavings before and after
# Create contingency table
table_shavings <- table(data$shavings.present.B, data$shavings.present.A)
# Print the table
print(table_shavings)
N Y
N 1 3
Y 4 10# Perform McNemar's test
mcnemar.test(table_shavings) # No significant difference in shavings present before and after treatment
McNemar's Chi-squared test with continuity correction
data: table_shavings
McNemar's chi-squared = 0, df = 1, p-value = 19) Comparing chew rates before and after dentist
t.test(data$chew.min.B, data$chew.min.A, paired = TRUE) # There is no evidence that dental treatment significantly changed the horses’ chew rate (chews per minute).
Paired t-test
data: data$chew.min.B and data$chew.min.A
t = 0.18898, df = 17, p-value = 0.8523
alternative hypothesis: true mean difference is not equal to 0
95 percent confidence interval:
-2.258688 2.703132
sample estimates:
mean difference
0.2222222 10) Comparing chew rates between age groups
Before dentist:
t.test(older_horses$chew.min.B, younger_horses$chew.min.B, paired = FALSE) # # No signif diff in chew rates between younger and older horses before treatment
Welch Two Sample t-test
data: older_horses$chew.min.B and younger_horses$chew.min.B
t = -0.10408, df = 12.08, p-value = 0.9188
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
-7.305926 6.639259
sample estimates:
mean of x mean of y
69.88889 70.22222 After dentist:
t.test(older_horses$chew.min.A, younger_horses$chew.min.A, paired = FALSE) # No signif diff in chew rates between younger and older horses after treatment
Welch Two Sample t-test
data: older_horses$chew.min.A and younger_horses$chew.min.A
t = 0.65041, df = 13.15, p-value = 0.5266
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
-3.347830 6.236719
sample estimates:
mean of x mean of y
70.55556 69.11111 11)
Older horses:
t.test(older_horses$chew.min.B,
older_horses$chew.min.A,
paired = TRUE) # No signif diff in chew rates of older horses before and after dentist
Paired t-test
data: older_horses$chew.min.B and older_horses$chew.min.A
t = -0.71842, df = 8, p-value = 0.4929
alternative hypothesis: true mean difference is not equal to 0
95 percent confidence interval:
-2.806548 1.473215
sample estimates:
mean difference
-0.6666667 Younger horses:
t.test(younger_horses$chew.min.B,
younger_horses$chew.min.A,
paired = TRUE) # No signif diff in chew rates of older horses before and after dentist
Paired t-test
data: younger_horses$chew.min.B and younger_horses$chew.min.A
t = 0.50621, df = 8, p-value = 0.6264
alternative hypothesis: true mean difference is not equal to 0
95 percent confidence interval:
-3.950503 6.172725
sample estimates:
mean difference
1.111111