Five ways to calculate a paired t-test

library(survey)
library(lme4)
library(lmerTest)

sleep_wide <- reshape(sleep, direction = "wide", idvar = "ID", timevar = "group")

Paired t-test

t.test(sleep_wide$extra.1, sleep_wide$extra.2, paired = TRUE)

    Paired t-test

data:  sleep_wide$extra.1 and sleep_wide$extra.2
t = -4.0621, df = 9, p-value = 0.002833
alternative hypothesis: true mean difference is not equal to 0
95 percent confidence interval:
 -2.4598858 -0.7001142
sample estimates:
mean difference 
          -1.58 

Intercept-only regression model of paired differences

lm(I(extra.1 - extra.2) ~ 1, data = sleep_wide) |>
  summary()

Call:
lm(formula = I(extra.1 - extra.2) ~ 1, data = sleep_wide)

Residuals:
   Min     1Q Median     3Q    Max 
 -3.02  -0.12   0.28   0.53   1.58 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)   
(Intercept)   -1.580      0.389  -4.062  0.00283 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.23 on 9 degrees of freedom

Unit fixed-effects regression

lm(extra ~ I(as.numeric(group == 1)) + ID + 0, data = sleep) |>
  summary()

Call:
lm(formula = extra ~ I(as.numeric(group == 1)) + ID + 0, data = sleep)

Residuals:
   Min     1Q Median     3Q    Max 
-1.510 -0.215  0.000  0.215  1.510 

Coefficients:
                          Estimate Std. Error t value Pr(>|t|)    
I(as.numeric(group == 1))   -1.580      0.389  -4.062 0.002833 ** 
ID1                          2.090      0.645   3.240 0.010155 *  
ID2                          0.390      0.645   0.605 0.560353    
ID3                          1.240      0.645   1.922 0.086717 .  
ID4                          0.240      0.645   0.372 0.718440    
ID5                          0.690      0.645   1.070 0.312585    
ID6                          4.690      0.645   7.271 4.71e-05 ***
ID7                          5.390      0.645   8.356 1.56e-05 ***
ID8                          1.990      0.645   3.085 0.013030 *  
ID9                          3.090      0.645   4.791 0.000987 ***
ID10                         3.490      0.645   5.411 0.000427 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.8697 on 9 degrees of freedom
Multiple R-squared:  0.9454,    Adjusted R-squared:  0.8788 
F-statistic: 14.18 on 11 and 9 DF,  p-value: 0.0002232

Survey design clustered by ID with equal weights

sleep_svy <- svydesign(ids = ~ ID, data = sleep, weights = 1)
sleep_svy

1 - level Cluster Sampling design (with replacement)
With (10) clusters.
svydesign(ids = ~ID, data = sleep, weights = 1)

svyglm(extra ~ I(as.numeric(group == 1)), design = sleep_svy) |>
  summary(df.resid = degf(sleep_svy))

Call:
svyglm(formula = extra ~ I(as.numeric(group == 1)), design = sleep_svy)

Survey design:
svydesign(ids = ~ID, data = sleep, weights = 1)

Coefficients:
                          Estimate Std. Error t value Pr(>|t|)   
(Intercept)                 2.3300     0.6332   3.680  0.00508 **
I(as.numeric(group == 1))  -1.5800     0.3890  -4.062  0.00283 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for gaussian family taken to be 3.415053)

Number of Fisher Scoring iterations: 2

Mixed model with random intercept by ID and Satterthwaite’s t

lmerTest::lmer(extra ~ I(as.numeric(group == 1)) + (1|ID), data = sleep) |>
  summary()

Linear mixed model fit by REML. t-tests use Satterthwaite's method ['lmerModLmerTest']
Formula: extra ~ I(as.numeric(group == 1)) + (1 | ID)
   Data: sleep

REML criterion at convergence: 70

Scaled residuals: 
     Min       1Q   Median       3Q      Max 
-1.63372 -0.34157  0.03346  0.31511  1.83859 

Random effects:
 Groups   Name        Variance Std.Dev.
 ID       (Intercept) 2.8483   1.6877  
 Residual             0.7564   0.8697  
Number of obs: 20, groups:  ID, 10

Fixed effects:
                          Estimate Std. Error      df t value Pr(>|t|)   
(Intercept)                 2.3300     0.6004 11.0814   3.881  0.00253 **
I(as.numeric(group == 1))  -1.5800     0.3890  9.0000  -4.062  0.00283 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Correlation of Fixed Effects:
            (Intr)
I(s.n(==1)) -0.324