One Factor Parametric Tests - R#

Independent-Samples t-test#

Samples: 1
Levels: 2
Between or Within Subjects: Between
Reporting: “The mean of ‘a’ was 14.63 (SD = 2.13) and of ‘b’ was 11.01 (SD = 1.75). This difference was statistically significant according to an independent-samples t-test (t(58) = 7.18, p < .0001).”

# Example data
# df has subjects (S), one between-Ss factor (X) w/levels (a,b), and continuous response (Y)
df <- read.csv("data/1F2LBs.csv")
head(df, 20)

A data.frame: 20 × 3
	S	X	Y
	<int>	<chr>	<dbl>
1	1	a	13.211290
2	2	b	9.376966
3	3	a	9.832110
4	4	b	13.241823
5	5	a	15.290763
6	6	b	11.926719
7	7	a	16.513046
8	8	b	11.817017
9	9	a	14.405615
10	10	b	7.186540
11	11	a	14.620504
12	12	b	9.200230
13	13	a	13.619696
14	14	b	8.989190
15	15	a	10.055274
16	16	b	12.827598
17	17	a	16.136434
18	18	b	11.604144
19	19	a	12.451788
20	20	b	11.255572

df$S = factor(df$S) # Subject id is nominal (unused)
df$X = factor(df$X) # X is a 2-level factor
t.test(Y ~ X, data=df, var.equal=TRUE) # use var.equal=FALSE if heteroscedastistic

	Two Sample t-test

data:  Y by X
t = 7.1775, df = 58, p-value = 1.475e-09
alternative hypothesis: true difference in means between group a and group b is not equal to 0
95 percent confidence interval:
 2.609128 4.627288
sample estimates:
mean in group a mean in group b 
       14.62769        11.00948 

Paired-Samples t-test#

Samples: 1
Levels: 2
Between or Within Subjects: Within
Reporting: “The mean of ‘a’ was 13.15 (SD = 2.53) and of ‘b’ was 14.37 (SD = 2.16). This difference was statistically significant according to a paired-samples t-test (t(29) = -2.14, p < .05).”

# Example data
# df has subjects (S), one within-Ss factor (X) w/levels (a,b), and continuous response (Y)
df <- read.csv("data/1F2LWs.csv")
head(df, 20)

A data.frame: 20 × 3
	S	X	Y
	<int>	<chr>	<dbl>
1	1	a	9.348176
2	1	b	16.280812
3	2	a	12.797245
4	2	b	14.638421
5	3	a	11.757036
6	3	b	14.221410
7	4	a	13.037475
8	4	b	15.172092
9	5	a	14.934822
10	5	b	16.080187
11	6	a	10.762074
12	6	b	11.558915
13	7	a	14.699569
14	7	b	15.364241
15	8	a	12.476783
16	8	b	13.468385
17	9	a	12.761169
18	9	b	17.316534
19	10	a	16.532902
20	10	b	16.405975

library(reshape2) # for dcast
df$S = factor(df$S) # Subject id is nominal
df$X = factor(df$X) # X is a 2-level factor
df2 <- dcast(df, S ~ X, value.var="Y") # make wide-format table
t.test(df2$a, df2$b, paired=TRUE) # homoscedasticity is irrelevant for a paired-samples t-test

	Paired t-test

data:  df2$a and df2$b
t = -2.1363, df = 29, p-value = 0.04123
alternative hypothesis: true mean difference is not equal to 0
95 percent confidence interval:
 -2.39877224 -0.05222605
sample estimates:
mean difference 
      -1.225499 

One-Way ANOVA#

Samples: 1
Levels: ≥2
Between or Within Subjects: Between
Reporting: “The mean of ‘a’ was 13.74 (SD = 2.84), of ‘b’ was 14.15 (SD = 2.65), and of ‘c’ was 9.08 (SD = 4.29). These differences were statistically significant according to a one-way ANOVA (F(2, 57) = 14.18, p < .0001).”

# Example data
# df has subjects (S), one between-Ss factor (X) w/levels (a,b,c), and continuous response (Y)
df <- read.csv("data/1F3LBs.csv")
head(df, 20)

A data.frame: 20 × 3
	S	X	Y
	<int>	<chr>	<dbl>
1	1	a	14.310439
2	2	b	17.390453
3	3	c	12.501365
4	4	a	17.943734
5	5	b	10.597671
6	6	c	9.652177
7	7	a	10.095838
8	8	b	15.324131
9	9	c	7.649627
10	10	a	13.517695
11	11	b	13.702848
12	12	c	19.033070
13	13	a	11.871676
14	14	b	12.177908
15	15	c	7.713374
16	16	a	11.698955
17	17	b	13.716288
18	18	c	9.492661
19	19	a	17.384638
20	20	b	13.506336

df$S = factor(df$S) # Subject id is nominal (unused)
df$X = factor(df$X) # X is a 3-level factor
m = aov(Y ~ X, data=df) # fit model
anova(m)

A anova: 2 × 5
	Df	Sum Sq	Mean Sq	F value	Pr(>F)
	<int>	<dbl>	<dbl>	<dbl>	<dbl>
X	2	316.8775	158.43873	14.18067	1.00343e-05
Residuals	57	636.8532	11.17286	NA	NA

One-Way Repeated Measures ANOVA#

Samples: 1
Levels: ≥2
Between or Within Subjects: Within
Reporting: “The mean of ‘a’ was 14.04 (SD = 2.98), of ‘b’ was 11.95 (SD = 1.98), and of ‘c’ was 11.40 (SD = 2.75). Mauchly’s test of sphericity indicated no sphericity violation (W = .926, p = .499), allowing for an uncorrected repeated measures ANOVA, which showed statistically significant differences (F(2, 38) = 6.57, p < .01).”

# Example data
# df has subjects (S), one within-Ss factor (X) w/levels (a,b,c), and continuous response (Y)
df <- read.csv("data/1F3LWs.csv")
head(df, 20)

A data.frame: 20 × 3
	S	X	Y
	<int>	<chr>	<dbl>
1	1	a	8.833569
2	1	b	11.849632
3	1	c	7.909041
4	2	a	15.361964
5	2	b	10.177478
6	2	c	13.245308
7	3	a	10.820555
8	3	b	9.240199
9	3	c	14.307353
10	4	a	13.089309
11	4	b	13.276904
12	4	c	11.683161
13	5	a	12.229721
14	5	b	10.442200
15	5	c	7.425539
16	6	a	15.471203
17	6	b	10.401901
18	6	c	14.725702
19	7	a	14.398177
20	7	b	10.755235

library(ez) # for ezANOVA

df$S = factor(df$S) # Subject id is nominal
df$X = factor(df$X) # X is a 3-level factor

m = ezANOVA(dv=Y, within=c(X), wid=S, type=3, data=df) # use c() for >1 factors

m$Mauchly # p<.05 indicates a sphericity violation

A data.frame: 1 × 4
	Effect	W	p	p<.05
	<chr>	<dbl>	<dbl>	<chr>
2	X	0.9256272	0.4987981

m$ANOVA # use if no violation

A data.frame: 1 × 7
	Effect	DFn	DFd	F	p	p<.05	ges
	<chr>	<dbl>	<dbl>	<dbl>	<dbl>	<chr>	<dbl>
2	X	2	38	6.569937	0.003543683	*	0.1682495

# if there is a sphericity violation, report the Greenhouse-Geisser or Huynh-Feldt correction
p = match(m$Sphericity$Effect, m$ANOVA$Effect) # positions of within-Ss effects in m$ANOVA
m$Sphericity$GGe.DFn = m$Sphericity$GGe * m$ANOVA$DFn[p] # Greenhouse-Geisser DFs
m$Sphericity$GGe.DFd = m$Sphericity$GGe * m$ANOVA$DFd[p]
m$Sphericity$HFe.DFn = m$Sphericity$HFe * m$ANOVA$DFn[p] # Huynh-Feldt DFs
m$Sphericity$HFe.DFd = m$Sphericity$HFe * m$ANOVA$DFd[p]
m$Sphericity # show results

A data.frame: 1 × 11
	Effect	GGe	p[GG]	p[GG]<.05	HFe	p[HF]	p[HF]<.05	GGe.DFn	GGe.DFd	HFe.DFn	HFe.DFd
	<chr>	<dbl>	<dbl>	<chr>	<dbl>	<dbl>	<chr>	<dbl>	<dbl>	<dbl>	<dbl>
2	X	0.9307756	0.00446249	*	1.027836	0.003543683	*	1.861551	35.36947	2.055672	39.05776

# the following also performs the equivalent repeated measures ANOVA, but does not address sphericity
m = aov(Y ~ X + Error(S/X), data=df)
summary(m)

Error: S
          Df Sum Sq Mean Sq F value Pr(>F)
Residuals 19  160.3   8.436               

Error: S:X
          Df Sum Sq Mean Sq F value  Pr(>F)   
X          2  78.12   39.06    6.57 0.00354 **
Residuals 38 225.93    5.95                   
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Statistical Analysis and Reporting

One Factor Parametric Tests - R

Contents

One Factor Parametric Tests - R#

Independent-Samples t-test#

Paired-Samples t-test#

One-Way ANOVA#

One-Way Repeated Measures ANOVA#