29.2: Comparación de dos medias (Sección 28.2)
- Page ID
- 150662
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Para comparar dos medias de muestras independientes, podemos usar la prueba t de dos muestras. Digamos que queremos comparar la presión arterial de fumadores y no fumadores; no tenemos expectativas para la dirección, por lo que usaremos una prueba de dos caras. Primero vamos a realizar un análisis de potencia, de nuevo para un pequeño efecto:
power_results_2sample <- pwr.t.test(d=0.2, power=0.8,
type='two.sample'
)
power_results_2sample##
## Two-sample t test power calculation
##
## n = 393
## d = 0.2
## sig.level = 0.05
## power = 0.8
## alternative = two.sided
##
## NOTE: n is number in *each* groupEsto nos dice que necesitamos 394 sujetos en cada grupo, así que vamos a muestrear 394 fumadores y 394 no fumadores del conjunto de datos de NHANES, y luego ponerlos en un solo marco de datos con una variable que denota su estado de tabaquismo.
nonsmoker_df <- NHANES_adult %>%
dplyr::filter(SmokeNow=="Yes") %>%
drop_na(BPSysAve) %>%
dplyr::select(BPSysAve,SmokeNow) %>%
sample_n(power_results_2sample$n)
smoker_df <- NHANES_adult %>%
dplyr::filter(SmokeNow=="No") %>%
drop_na(BPSysAve) %>%
dplyr::select(BPSysAve,SmokeNow) %>%
sample_n(power_results_2sample$n)
sample_df <- smoker_df %>%
bind_rows(nonsmoker_df)Probemos nuestra hipótesis usando una prueba t estándar de dos muestras. Podemos usar la notación de fórmula para especificar el análisis, tal como lo haríamos para lm ().
t.test(BPSysAve ~ SmokeNow, data=sample_df)##
## Welch Two Sample t-test
##
## data: BPSysAve by SmokeNow
## t = 4, df = 775, p-value = 3e-05
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## 2.9 7.8
## sample estimates:
## mean in group No mean in group Yes
## 125 120Esto nos demuestra que hay una diferencia significativa, aunque la dirección es sorprendente: ¡Los fumadores tienen presión arterial más baja!
Veamos el factor Bayes para cuantificar la evidencia:
sample_df <- sample_df %>%
mutate(SmokeNowInt=as.integer(SmokeNow))
ttestBF(formula=BPSysAve ~ SmokeNowInt,
data=sample_df)## Bayes factor analysis
## --------------
## [1] Alt., r=0.707 : 440 ±0%
##
## Against denominator:
## Null, mu1-mu2 = 0
## ---
## Bayes factor type: BFindepSample, JZSEsto demuestra que existe evidencia muy contundente contra la hipótesis nula de no diferencia.