For this tutorial, I am using my dissertation data. I am using one speaker’s data so we don’t have to account for individual differences, and I am only using the midpoint of the vowel.
normtimeBP02 = normtimedata %>% filter(Speaker =="BP02", NormTime==0.5) %>% mutate(Vowel = as.factor(Vowel), Nasality = as.factor(Nasality))
head(normtimeBP02)## Speaker Word Vowel Nasality RepNo NormTime Type label Time F1
## 1 BP02 abunda u nasal 1 0.5 clean un 3.482666 415.2112
## 2 BP02 abunda u nasal 2 0.5 clean un 5.903856 367.2511
## 3 BP02 abunda u nasal 3 0.5 clean un 8.508366 314.8519
## 4 BP02 abunda u nasal 4 0.5 clean un 11.229561 301.0646
## 5 BP02 abunda u nasal 5 0.5 clean un 14.017535 335.9188
## 6 BP02 abunda u nasal 6 0.5 clean un 16.364416 310.0842
## F2 F3 F2_3
## 1 2247.165 2892.599 322.7169
## 2 2300.860 2897.852 298.4962
## 3 1507.593 2549.471 520.9387
## 4 1368.582 2373.538 502.4779
## 5 1251.865 2441.061 594.5982
## 6 2077.428 2695.539 309.0552A contrast is a linear combination of variables that allows comparison of different treatments. Categorical variables are entered into a regression analysis as a sequence of \(n-1\) variables. Generally, these are dummy variables. There are four built-in contrast coding schemes in R. You can also build your own contrast coding scheme. No matter which coding scheme you use, you will always have \(n−1\) columns for \(n\) levels of a variable in your contrast coding scheme.
Different contrasts can help answer different research questions. For example, sometimes it is not terribly informative to compare groups to a reference level. You might want to make a comparison between groups and the overall trend in your data. With ordinal data, the distance between groups might not be the same between subsequent groups. Therefore it would make more sense to make stepwise comparisons.
By default, a variable does not have a contrast set in place - R will default to the dummy coding system when running a linear regression. In order to set a contrast in R, you can use the contr._X_() function for treatment, sum, and Helmert contrasts, or define any contrast manually. Be aware that this changes your dataset. You might want to consider creating a new variable as a copy of your original one, and set the contrasts on that variable. Alternatively, set the contrast back to ‘treatment’ when you’re done your analysis. Note that there is no inherent problem with leaving a contrast set, except it might surprise you when you run a different analysis on your data!
Make sure to set all variables’ contrasts before starting.
The coding scheme you are most familiar with is dummy coding, or treatment coding. It compares each level of a categorical variable to a reference level. By default, the reference level is the first level of the categorical variable, in alphabetical order.
The contrast matrix for a categorical variable with four levels looks like this:
contr.treatment(4)## 2 3 4
## 1 0 0 0
## 2 1 0 0
## 3 0 1 0
## 4 0 0 1contrasts(normtimeBP02$Nasality)## nasal_final nasalized oral
## nasal 0 0 0
## nasal_final 1 0 0
## nasalized 0 1 0
## oral 0 0 1contrasts(normtimeBP02$Vowel)## i u
## a 0 0
## i 1 0
## u 0 1#if I wanted to change the contrast from a different setting to dummy coding:
contrasts(normtimeBP02$Nasality) = contr.treatment(4)
nasal_lm1 = lm(F1~Vowel + Nasality, data = normtimeBP02)
summary(nasal_lm1)##
## Call:
## lm(formula = F1 ~ Vowel + Nasality, data = normtimeBP02)
##
## Residuals:
## Min 1Q Median 3Q Max
## -282.70 -59.22 -7.81 61.11 741.99
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 440.51 11.90 37.023 < 2e-16 ***
## Voweli -239.44 11.96 -20.017 < 2e-16 ***
## Vowelu -158.14 11.87 -13.318 < 2e-16 ***
## Nasality2 89.20 13.65 6.535 1.94e-10 ***
## Nasality3 124.26 13.72 9.059 < 2e-16 ***
## Nasality4 147.61 13.86 10.653 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 97.71 on 399 degrees of freedom
## Multiple R-squared: 0.575, Adjusted R-squared: 0.5697
## F-statistic: 108 on 5 and 399 DF, p-value: < 2.2e-16Here, we are comparing each level of a factor to a reference level. In this case, the reference level is /a/ for Vowel, and nasal for Nasality. These are chosen because they are the first alphabetically.
In order to change the reference level, you can either change the contrasts, or relevel the factor.
#Change the contrast with a different base (i.e., reference) level
contrasts(normtimeBP02$Nasality) = contr.treatment(4, base = 3)
contrasts(normtimeBP02$Nasality)## 1 2 4
## nasal 1 0 0
## nasal_final 0 1 0
## nasalized 0 0 0
## oral 0 0 1nasal_lm2 = lm(F1~Vowel + Nasality, data = normtimeBP02)
summary(nasal_lm2)##
## Call:
## lm(formula = F1 ~ Vowel + Nasality, data = normtimeBP02)
##
## Residuals:
## Min 1Q Median 3Q Max
## -282.70 -59.22 -7.81 61.11 741.99
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 564.77 11.94 47.306 <2e-16 ***
## Voweli -239.44 11.96 -20.017 <2e-16 ***
## Vowelu -158.14 11.87 -13.318 <2e-16 ***
## Nasality1 -124.26 13.72 -9.059 <2e-16 ***
## Nasality2 -35.06 13.62 -2.575 0.0104 *
## Nasality4 23.35 13.82 1.689 0.0920 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 97.71 on 399 degrees of freedom
## Multiple R-squared: 0.575, Adjusted R-squared: 0.5697
## F-statistic: 108 on 5 and 399 DF, p-value: < 2.2e-16#Change the contrast by releveling the factor
normtimeBP02$Nasality = relevel(normtimeBP02$Nasality, ref = "nasalized")
contrasts(normtimeBP02$Nasality)## nasal nasal_final oral
## nasalized 0 0 0
## nasal 1 0 0
## nasal_final 0 1 0
## oral 0 0 1nasal_lm3 = lm(F1~Vowel + Nasality, data = normtimeBP02)
summary(nasal_lm3)##
## Call:
## lm(formula = F1 ~ Vowel + Nasality, data = normtimeBP02)
##
## Residuals:
## Min 1Q Median 3Q Max
## -282.70 -59.22 -7.81 61.11 741.99
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 564.77 11.94 47.306 <2e-16 ***
## Voweli -239.44 11.96 -20.017 <2e-16 ***
## Vowelu -158.14 11.87 -13.318 <2e-16 ***
## Nasalitynasal -124.26 13.72 -9.059 <2e-16 ***
## Nasalitynasal_final -35.06 13.62 -2.575 0.0104 *
## Nasalityoral 23.35 13.82 1.689 0.0920 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 97.71 on 399 degrees of freedom
## Multiple R-squared: 0.575, Adjusted R-squared: 0.5697
## F-statistic: 108 on 5 and 399 DF, p-value: < 2.2e-16Other coding methods allow the researcher to ask specific questions about the relationship between variables. Basic contrast coding is sometimes called simple coding, though it can be difficult to wrap your head around.
Contrast coding in this case takes the grand mean as the intercept, and then each level of a factor is changed to be compared to a reference level.
In the example below, group 4 is the reference group and the first comparison compares group 1 to group 4, the second comparison compares group 2 to group 4, and the third comparison compares group 3 to group 4.
c<-contr.treatment(4)
my.coding<-matrix(rep(1/4, 12), ncol=3)
my.simple<-c-my.coding
my.simple## 2 3 4
## 1 -0.25 -0.25 -0.25
## 2 0.75 -0.25 -0.25
## 3 -0.25 0.75 -0.25
## 4 -0.25 -0.25 0.75c2<-contr.treatment(3)
my.coding2<-matrix(rep(1/3, 6), ncol=2)
my.simple2<-c2-my.coding2
my.simple2## 2 3
## 1 -0.3333333 -0.3333333
## 2 0.6666667 -0.3333333
## 3 -0.3333333 0.6666667contrasts(normtimeBP02$Nasality) = my.simple
contrasts(normtimeBP02$Vowel) = my.simple2
nasal_lm4 = lm(F1~Vowel + Nasality, data = normtimeBP02)
summary(nasal_lm4)##
## Call:
## lm(formula = F1 ~ Vowel + Nasality, data = normtimeBP02)
##
## Residuals:
## Min 1Q Median 3Q Max
## -282.70 -59.22 -7.81 61.11 741.99
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 398.253 4.857 81.999 <2e-16 ***
## Vowel2 -239.437 11.962 -20.017 <2e-16 ***
## Vowel3 -158.137 11.874 -13.318 <2e-16 ***
## Nasality2 -124.261 13.717 -9.059 <2e-16 ***
## Nasality3 -35.059 13.617 -2.575 0.0104 *
## Nasality4 23.346 13.823 1.689 0.0920 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 97.71 on 399 degrees of freedom
## Multiple R-squared: 0.575, Adjusted R-squared: 0.5697
## F-statistic: 108 on 5 and 399 DF, p-value: < 2.2e-16Here, vowel2 is the difference between /a/ and /u/, and vowel3 is the difference between /i/ and /u/.
In sum coding, we compare the mean of a dependent variable for a given level to the overall mean of the dependent variable.
This grand mean is not the mean of the dependent variable. Rather, it is the mean of means of the dependent variable at each level of the categorical variable. This does not take into account different group sizes.
For example, the first column given below compares variable 1 to the grand mean, the second compares variable 2 to the grand mean, the third compares variable 3 to the grand mean.
contr.sum(4)## [,1] [,2] [,3]
## 1 1 0 0
## 2 0 1 0
## 3 0 0 1
## 4 -1 -1 -1contrasts(normtimeBP02$Nasality) = contr.sum(4)
contrasts(normtimeBP02$Vowel) =contr.sum(3)
nasal_lm5 = lm(F1~Vowel + Nasality, data = normtimeBP02)
summary(nasal_lm5)##
## Call:
## lm(formula = F1 ~ Vowel + Nasality, data = normtimeBP02)
##
## Residuals:
## Min 1Q Median 3Q Max
## -282.70 -59.22 -7.81 61.11 741.99
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 398.253 4.857 81.999 < 2e-16 ***
## Vowel1 132.525 6.894 19.225 < 2e-16 ***
## Vowel2 -106.913 6.881 -15.538 < 2e-16 ***
## Nasality1 33.993 8.390 4.052 6.12e-05 ***
## Nasality2 -90.268 8.417 -10.724 < 2e-16 ***
## Nasality3 -1.065 8.336 -0.128 0.898
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 97.71 on 399 degrees of freedom
## Multiple R-squared: 0.575, Adjusted R-squared: 0.5697
## F-statistic: 108 on 5 and 399 DF, p-value: < 2.2e-16In order to get the mean values for /u/ and ’oral vowels (aka the last levels of each contrast), we need to subtract all of the regression coefficients from the grand mean.
/u/ = intercept - Vowel1 – Vowel2 = 398.253 - 132.525 – (-106.913) = 372.6416 Oral = intercept – Nasality1 – nasality2 –nasality3 = 398.253 - 33.993 – (- 90.267)- (-1.06) = 455.5928
Helmert coding compares each level of a categorical variable to the mean of subsequent levels of the variable.
For the example below, Helmert coding would output the first regression coefficient as the mean of level 1 compared to the mean of levels 2, 3, and 4. The second regression coefficient would be the mean of level 2 compared to the mean of levels 3 and 4. The third regression coefficient would be the mean of level 3 minus the mean of level 4.
Note that we need to define the contrasts ourselves here, as the contr.helmert() function actually employs reverse Helmert coding (below).
myhelmert = matrix(c(3/4, -1/4, -1/4, -1/4, 0, 2/3, -1/3, -1/3, 0, 0, 1/
2, -1/2), ncol = 3)
myhelmert## [,1] [,2] [,3]
## [1,] 0.75 0.0000000 0.0
## [2,] -0.25 0.6666667 0.0
## [3,] -0.25 -0.3333333 0.5
## [4,] -0.25 -0.3333333 -0.5myhelmert2 = matrix(c(2/3, -1/3, -1/3, 0, .5, -.5), ncol = 2)
myhelmert2## [,1] [,2]
## [1,] 0.6666667 0.0
## [2,] -0.3333333 0.5
## [3,] -0.3333333 -0.5contrasts(normtimeBP02$Nasality) = myhelmert
contrasts(normtimeBP02$Vowel) = myhelmert2
nasal_lm6 = lm(F1~Vowel + Nasality, data = normtimeBP02)
summary(nasal_lm6)##
## Call:
## lm(formula = F1 ~ Vowel + Nasality, data = normtimeBP02)
##
## Residuals:
## Min 1Q Median 3Q Max
## -282.70 -59.22 -7.81 61.11 741.99
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 398.253 4.857 81.999 < 2e-16 ***
## Vowel1 198.787 10.340 19.225 < 2e-16 ***
## Vowel2 -81.301 11.852 -6.860 2.64e-11 ***
## Nasality1 45.325 11.187 4.052 6.12e-05 ***
## Nasality2 -118.405 11.910 -9.942 < 2e-16 ***
## Nasality3 -58.405 13.757 -4.245 2.72e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 97.71 on 399 degrees of freedom
## Multiple R-squared: 0.575, Adjusted R-squared: 0.5697
## F-statistic: 108 on 5 and 399 DF, p-value: < 2.2e-16Reverse Helmert coding compares each level of a categorical variable to the mean of previous levels of the variable.
In the example below, Reverse Helmert coding outputs the first regression coefficient as the mean of level 2 compared to the mean of level 1. The second regression coefficient would be the mean of level 3 compared to the mean of levels 1 and 2. The third regression coefficient would be the mean of level 4 minus the mean of levels 1, 2, and 3.
contr.helmert(4)## [,1] [,2] [,3]
## 1 -1 -1 -1
## 2 1 -1 -1
## 3 0 2 -1
## 4 0 0 3myrevhelmert = matrix(c(-1/2, 1/2, 0, 0, -1/3, -1/3, 2/3, 0, -1/4, -1/4,
-1/4, 3/4), ncol = 3)
myrevhelmert2 = matrix(c(-.5, .5, 0, -1/3, -1/3, 2/3), ncol = 2)
myrevhelmert## [,1] [,2] [,3]
## [1,] -0.5 -0.3333333 -0.25
## [2,] 0.5 -0.3333333 -0.25
## [3,] 0.0 0.6666667 -0.25
## [4,] 0.0 0.0000000 0.75myrevhelmert2## [,1] [,2]
## [1,] -0.5 -0.3333333
## [2,] 0.5 -0.3333333
## [3,] 0.0 0.6666667contr.helmert(4)## [,1] [,2] [,3]
## 1 -1 -1 -1
## 2 1 -1 -1
## 3 0 2 -1
## 4 0 0 3contr.helmert(3)## [,1] [,2]
## 1 -1 -1
## 2 1 -1
## 3 0 2contrasts(normtimeBP02$Nasality) = myrevhelmert
contrasts(normtimeBP02$Vowel) = myrevhelmert2
nasal_lm7 = lm(F1~Vowel + Nasality, data = normtimeBP02)
summary(nasal_lm7)##
## Call:
## lm(formula = F1 ~ Vowel + Nasality, data = normtimeBP02)
##
## Residuals:
## Min 1Q Median 3Q Max
## -282.70 -59.22 -7.81 61.11 741.99
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 398.253 4.857 81.999 < 2e-16 ***
## Vowel1 -239.437 11.962 -20.017 < 2e-16 ***
## Vowel2 -38.418 10.245 -3.750 0.000203 ***
## Nasality1 -124.261 13.717 -9.059 < 2e-16 ***
## Nasality2 27.072 11.783 2.298 0.022101 *
## Nasality3 76.453 11.339 6.743 5.47e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 97.71 on 399 degrees of freedom
## Multiple R-squared: 0.575, Adjusted R-squared: 0.5697
## F-statistic: 108 on 5 and 399 DF, p-value: < 2.2e-16contrasts(normtimeBP02$Nasality) = contr.helmert(4)
contrasts(normtimeBP02$Vowel) = contr.helmert(3)
nasal_lm8 = lm(F1~Vowel + Nasality, data = normtimeBP02)
summary(nasal_lm8)##
## Call:
## lm(formula = F1 ~ Vowel + Nasality, data = normtimeBP02)
##
## Residuals:
## Min 1Q Median 3Q Max
## -282.70 -59.22 -7.81 61.11 741.99
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 398.253 4.857 81.999 < 2e-16 ***
## Vowel1 -119.719 5.981 -20.017 < 2e-16 ***
## Vowel2 -12.806 3.415 -3.750 0.000203 ***
## Nasality1 -62.131 6.858 -9.059 < 2e-16 ***
## Nasality2 9.024 3.928 2.298 0.022101 *
## Nasality3 19.113 2.835 6.743 5.47e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 97.71 on 399 degrees of freedom
## Multiple R-squared: 0.575, Adjusted R-squared: 0.5697
## F-statistic: 108 on 5 and 399 DF, p-value: < 2.2e-16Notice that the contr.helmert() function has different weights to it. This is fine to use, you just have to scale your data back down.
One has fractional contrasts, and second one has whole number contrasts. They have the same p values and t values, the only difference is in magnitude of estimates and errors. Using the fractional one ensures that one unit change in contrast is the difference between categories. (Using contr.helmert() means that one unit change is only a fraction of the difference between categories.)
It is possible to define your own contrasts, based on your personal research questions. Note that you still need to include \(n-1\) columns in your contrast matrix. If you do not, R will try to supply the most appropriate one it thinks is possible, but the weights of the contrast matrix might be a bit wonky.
Another example of contrast coding would be, say we had three groups: English, Spanish, and French speakers, who participated in a lexical decision task. If we wanted to, we could compare between English and non-English (i.e., Spanish and French), and then between Spanish and French speakers. The contrast for that would look like this:
matrix(c(1, -1/2, -1/2, 0, .5, -.5), ncol = 2)## [,1] [,2]
## [1,] 1.0 0.0
## [2,] -0.5 0.5
## [3,] -0.5 -0.5The intercept would be the grand mean. The first coefficient the model would produce would be the difference between the intercept and English mean, and also would be twice the difference between the intercept and the average of Spanish and French means. The second coefficient would be the difference between the French and Spanish speakers, divided by 2.
mycontrasts = contrasts(normtimeBP02$Nasality)
mycontrasts[,1] = c(-1/3, -1/3, -1/3, 1)
mycontrasts[,2] = c(1, -1/2, -1/2, 0)
mycontrasts[,3] = c(0, .5, -.5, 0)
contrasts(normtimeBP02$Nasality) = mycontrasts
mycontrasts2 = contrasts(normtimeBP02$Vowel)
mycontrasts2[,1] = c(1, -1/2, -1/2)
mycontrasts2[,2] = c(0,.5,-.5)
contrasts(normtimeBP02$Vowel) = mycontrasts2
nasal_lm9 = lm(F1~Vowel + Nasality, data = normtimeBP02)
summary(nasal_lm9)##
## Call:
## lm(formula = F1 ~ Vowel + Nasality, data = normtimeBP02)
##
## Residuals:
## Min 1Q Median 3Q Max
## -282.70 -59.22 -7.81 61.11 741.99
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 398.253 4.857 81.999 < 2e-16 ***
## Vowel1 132.525 6.894 19.225 < 2e-16 ***
## Vowel2 -81.301 11.852 -6.860 2.64e-11 ***
## Nasality1 57.339 8.504 6.743 5.47e-11 ***
## Nasality2 53.107 7.894 6.728 6.00e-11 ***
## Nasality3 -89.202 13.650 -6.535 1.94e-10 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 97.71 on 399 degrees of freedom
## Multiple R-squared: 0.575, Adjusted R-squared: 0.5697
## F-statistic: 108 on 5 and 399 DF, p-value: < 2.2e-16What if we wanted to do comparisons between groups, beyond looking at coefficients of models? Estimated marginal means are a common way of looking at main effects and interactions, and can be used for both dummy coding and effects coding.
If you have used the lsmeans package in the past, you are probably familiar with emmeans. The two packages are very similar, just with some back-end changes that won’t affect your analysis.
library(emmeans)
contrasts(normtimeBP02$Nasality) = contr.treatment(4)
contrasts(normtimeBP02$Vowel) =contr.treatment(3)
nasal_lm1 = lm(F1~Vowel + Nasality, data = normtimeBP02)
summary(nasal_lm1)##
## Call:
## lm(formula = F1 ~ Vowel + Nasality, data = normtimeBP02)
##
## Residuals:
## Min 1Q Median 3Q Max
## -282.70 -59.22 -7.81 61.11 741.99
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 564.77 11.94 47.306 <2e-16 ***
## Vowel2 -239.44 11.96 -20.017 <2e-16 ***
## Vowel3 -158.14 11.87 -13.318 <2e-16 ***
## Nasality2 -124.26 13.72 -9.059 <2e-16 ***
## Nasality3 -35.06 13.62 -2.575 0.0104 *
## Nasality4 23.35 13.82 1.689 0.0920 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 97.71 on 399 degrees of freedom
## Multiple R-squared: 0.575, Adjusted R-squared: 0.5697
## F-statistic: 108 on 5 and 399 DF, p-value: < 2.2e-16emmeans(nasal_lm1, "Vowel", by = "Nasality")## Nasality = nasalized:
## Vowel emmean SE df lower.CL upper.CL
## a 565 11.9 399 541 588
## i 325 11.9 399 302 349
## u 407 11.7 399 384 430
##
## Nasality = nasal:
## Vowel emmean SE df lower.CL upper.CL
## a 441 11.9 399 417 464
## i 201 12.0 399 178 225
## u 282 11.9 399 259 306
##
## Nasality = nasal_final:
## Vowel emmean SE df lower.CL upper.CL
## a 530 11.8 399 507 553
## i 290 11.8 399 267 314
## u 372 11.7 399 348 395
##
## Nasality = oral:
## Vowel emmean SE df lower.CL upper.CL
## a 588 12.1 399 564 612
## i 349 12.0 399 325 372
## u 430 12.0 399 406 454
##
## Confidence level used: 0.95pairs(emmeans(nasal_lm1, "Vowel", by = "Nasality"))## Nasality = nasalized:
## contrast estimate SE df t.ratio p.value
## a - i 239.4 12.0 399 20.017 <.0001
## a - u 158.1 11.9 399 13.318 <.0001
## i - u -81.3 11.9 399 -6.860 <.0001
##
## Nasality = nasal:
## contrast estimate SE df t.ratio p.value
## a - i 239.4 12.0 399 20.017 <.0001
## a - u 158.1 11.9 399 13.318 <.0001
## i - u -81.3 11.9 399 -6.860 <.0001
##
## Nasality = nasal_final:
## contrast estimate SE df t.ratio p.value
## a - i 239.4 12.0 399 20.017 <.0001
## a - u 158.1 11.9 399 13.318 <.0001
## i - u -81.3 11.9 399 -6.860 <.0001
##
## Nasality = oral:
## contrast estimate SE df t.ratio p.value
## a - i 239.4 12.0 399 20.017 <.0001
## a - u 158.1 11.9 399 13.318 <.0001
## i - u -81.3 11.9 399 -6.860 <.0001
##
## P value adjustment: tukey method for comparing a family of 3 estimatescontrasts(normtimeBP02$Nasality) = contr.sum(4)
contrasts(normtimeBP02$Vowel) =contr.sum(3)
nasal_lm5 = lm(F1~Vowel + Nasality, data = normtimeBP02)
summary(nasal_lm5)##
## Call:
## lm(formula = F1 ~ Vowel + Nasality, data = normtimeBP02)
##
## Residuals:
## Min 1Q Median 3Q Max
## -282.70 -59.22 -7.81 61.11 741.99
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 398.253 4.857 81.999 < 2e-16 ***
## Vowel1 132.525 6.894 19.225 < 2e-16 ***
## Vowel2 -106.913 6.881 -15.538 < 2e-16 ***
## Nasality1 33.993 8.390 4.052 6.12e-05 ***
## Nasality2 -90.268 8.417 -10.724 < 2e-16 ***
## Nasality3 -1.065 8.336 -0.128 0.898
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 97.71 on 399 degrees of freedom
## Multiple R-squared: 0.575, Adjusted R-squared: 0.5697
## F-statistic: 108 on 5 and 399 DF, p-value: < 2.2e-16emmeans(nasal_lm5, "Vowel", by = "Nasality")## Nasality = nasalized:
## Vowel emmean SE df lower.CL upper.CL
## a 565 11.9 399 541 588
## i 325 11.9 399 302 349
## u 407 11.7 399 384 430
##
## Nasality = nasal:
## Vowel emmean SE df lower.CL upper.CL
## a 441 11.9 399 417 464
## i 201 12.0 399 178 225
## u 282 11.9 399 259 306
##
## Nasality = nasal_final:
## Vowel emmean SE df lower.CL upper.CL
## a 530 11.8 399 507 553
## i 290 11.8 399 267 314
## u 372 11.7 399 348 395
##
## Nasality = oral:
## Vowel emmean SE df lower.CL upper.CL
## a 588 12.1 399 564 612
## i 349 12.0 399 325 372
## u 430 12.0 399 406 454
##
## Confidence level used: 0.95pairs(emmeans(nasal_lm5, "Vowel", by = "Nasality"))## Nasality = nasalized:
## contrast estimate SE df t.ratio p.value
## a - i 239.4 12.0 399 20.017 <.0001
## a - u 158.1 11.9 399 13.318 <.0001
## i - u -81.3 11.9 399 -6.860 <.0001
##
## Nasality = nasal:
## contrast estimate SE df t.ratio p.value
## a - i 239.4 12.0 399 20.017 <.0001
## a - u 158.1 11.9 399 13.318 <.0001
## i - u -81.3 11.9 399 -6.860 <.0001
##
## Nasality = nasal_final:
## contrast estimate SE df t.ratio p.value
## a - i 239.4 12.0 399 20.017 <.0001
## a - u 158.1 11.9 399 13.318 <.0001
## i - u -81.3 11.9 399 -6.860 <.0001
##
## Nasality = oral:
## contrast estimate SE df t.ratio p.value
## a - i 239.4 12.0 399 20.017 <.0001
## a - u 158.1 11.9 399 13.318 <.0001
## i - u -81.3 11.9 399 -6.860 <.0001
##
## P value adjustment: tukey method for comparing a family of 3 estimates