Resumen

Este informe presenta un análisis del salto vertical (Counter Movement Jump o CMJ) a partir de datos experimentales.
El objetivo principal es identificar la variable con mayor correlación con la altura de salto, construir un modelo lineal simple y evaluar su ajuste y significancia estadística.
El documento incluye procedimientos de limpieza de datos, análisis exploratorio, correlaciones, modelamiento lineal, diagnóstico de supuestos, validación cruzada y una interpretación práctica de los resultados.

1. Librerías necesarias

required <- c("tidyverse", "readr", "janitor", "corrplot", "psych",
              "broom", "car", "caret", "lmtest", "gt")

to_install <- required[!required %in% installed.packages()[, "Package"]]
if(length(to_install)) install.packages(to_install, repos = "https://cloud.r-project.org")

lapply(required, library, character.only = TRUE)
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##  [7] "tidyr"     "tibble"    "ggplot2"   "tidyverse" "stats"     "graphics" 
## [13] "grDevices" "utils"     "datasets"  "methods"   "base"     
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## [13] "forcats"   "stringr"   "dplyr"     "purrr"     "readr"     "tidyr"    
## [19] "tibble"    "ggplot2"   "tidyverse" "stats"     "graphics"  "grDevices"
## [25] "utils"     "datasets"  "methods"   "base"

2. Lectura y limpieza de datos

ruta <- "C:/Users/monav/Downloads/Actividad 2 - c2_data (1).csv"

df <- read_csv(ruta, show_col_types = FALSE) %>%
  clean_names() %>%
  rename(jump_height_cm = jump_height_flight_time)

df$player_id <- as.factor(df$player_id)

head(df)

3. Selección de variables numéricas

num_vars <- df %>% select(where(is.numeric))
summary(num_vars)
##        x1        braking_phase_duration concentric_rfd  concentric_rfd_100ms
##  Min.   :  1.0   Min.   :0.1048         Min.   :    0   Min.   :-24166      
##  1st Qu.:121.2   1st Qu.:0.1955         1st Qu.:  731   1st Qu.: -4548      
##  Median :241.5   Median :0.2221         Median : 1630   Median : -2809      
##  Mean   :241.5   Mean   :0.2287         Mean   : 2269   Mean   : -3236      
##  3rd Qu.:361.8   3rd Qu.:0.2558         3rd Qu.: 3094   3rd Qu.: -1332      
##  Max.   :482.0   Max.   :0.4268         Max.   :17800   Max.   :  5077      
##  countermovement_depth eccentric_mean_power_bm eccentric_peak_force_bm
##  Min.   :-49.17        Min.   :3.279           Min.   :19.56          
##  1st Qu.:-31.96        1st Qu.:5.912           1st Qu.:26.48          
##  Median :-27.10        Median :6.577           Median :28.76          
##  Mean   :-27.71        Mean   :6.614           Mean   :29.40          
##  3rd Qu.:-23.55        3rd Qu.:7.267           3rd Qu.:31.51          
##  Max.   :-11.61        Max.   :9.486           Max.   :52.03          
##  force_at_zero_velocity jump_height_cm  peak_power_bm   positive_impulse
##  Min.   :1139           Min.   :24.42   Min.   :39.92   Min.   : 378.2  
##  1st Qu.:2175           1st Qu.:36.03   1st Qu.:55.79   1st Qu.: 559.6  
##  Median :2505           Median :41.06   Median :60.81   Median : 708.1  
##  Mean   :2511           Mean   :41.18   Mean   :61.20   Mean   : 694.5  
##  3rd Qu.:2829           3rd Qu.:46.24   3rd Qu.:66.51   3rd Qu.: 815.9  
##  Max.   :4582           Max.   :61.41   Max.   :92.66   Max.   :1231.6  
##    total_work    
##  Min.   : 428.8  
##  1st Qu.: 727.6  
##  Median : 906.4  
##  Mean   : 899.6  
##  3rd Qu.:1042.4  
##  Max.   :1821.5

4. Matriz de correlaciones

corr_mat <- cor(num_vars, use = "pairwise.complete.obs")

corrplot(corr_mat, method = "color", type = "upper",
         tl.cex = 0.8, addCoef.col = "black", number.cex = 0.6)

5. Correlaciones con la variable objetivo

cors_with_target <- tibble(
  variable = setdiff(names(num_vars), "jump_height_cm"),
  r = corr_mat[setdiff(names(num_vars), "jump_height_cm"), "jump_height_cm"]
) %>%
  arrange(desc(abs(r)))

print(cors_with_target)
## # A tibble: 11 × 2
##    variable                      r
##    <chr>                     <dbl>
##  1 positive_impulse         0.669 
##  2 peak_power_bm            0.658 
##  3 total_work               0.654 
##  4 eccentric_mean_power_bm  0.501 
##  5 x1                       0.498 
##  6 force_at_zero_velocity   0.410 
##  7 countermovement_depth   -0.346 
##  8 concentric_rfd          -0.189 
##  9 eccentric_peak_force_bm  0.136 
## 10 braking_phase_duration  -0.0547
## 11 concentric_rfd_100ms     0.0435
predictor <- cors_with_target %>% slice_max(abs(r), n = 1) %>% pull(variable)
cat("Variable predictora elegida automáticamente:", predictor, "\n")
## Variable predictora elegida automáticamente: positive_impulse

6. Modelo lineal simple

data_model <- df %>% select(jump_height_cm, all_of(predictor)) %>% drop_na()
f <- as.formula(paste("jump_height_cm ~", predictor))
model <- lm(f, data = data_model)

summary(model)
## 
## Call:
## lm(formula = f, data = data_model)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -17.2780  -3.5503  -0.3116   2.9595  15.2624 
## 
## Coefficients:
##                   Estimate Std. Error t value Pr(>|t|)    
## (Intercept)      19.865360   1.106595   17.95   <2e-16 ***
## positive_impulse  0.030696   0.001557   19.72   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5.179 on 480 degrees of freedom
## Multiple R-squared:  0.4475, Adjusted R-squared:  0.4464 
## F-statistic: 388.8 on 1 and 480 DF,  p-value: < 2.2e-16

7. Visualización de la relación

ggplot(data_model, aes_string(x = predictor, y = "jump_height_cm")) +
  geom_point(alpha = 0.6, color = "darkblue") +
  geom_smooth(method = "lm", se = TRUE, color = "red", linewidth = 1) +
  labs(title = paste("Relación entre", predictor, "y Jump Height"),
       x = predictor, y = "Altura de salto (cm)") +
  theme_minimal()

8. Diagnóstico del modelo

par(mfrow = c(2, 2))
plot(model)

par(mfrow = c(1, 1))

bp <- bptest(model)
sh <- shapiro.test(resid(model))

cat("\n=== DIAGNÓSTICO DEL MODELO ===\n")
## 
## === DIAGNÓSTICO DEL MODELO ===
if (bp$p.value < 0.05) {
  cat("⚠️ Heterocedasticidad detectada (p =", round(bp$p.value, 4), ")\n")
} else {
  cat("✅ Homocedasticidad cumplida (p =", round(bp$p.value, 4), ")\n")
}
## ⚠️ Heterocedasticidad detectada (p = 0 )
if (sh$p.value > 0.05) {
  cat("✅ Residuos normales (p =", round(sh$p.value, 4), ")\n")
} else {
  cat("⚠️ Residuos no normales (p =", round(sh$p.value, 4), ")\n")
}
## ⚠️ Residuos no normales (p = 0.0396 )

9. Validación cruzada (10 pliegues)

train_control <- trainControl(method = "cv", number = 10)
cv_model <- train(f, data = data_model, method = "lm", trControl = train_control)
cv_model$results

10. Resultados e interpretación

cat("\n=== RESULTADOS DEL MODELO ===\n")
## 
## === RESULTADOS DEL MODELO ===
cat("Variable dependiente: Jump Height (Flight Time)\n")
## Variable dependiente: Jump Height (Flight Time)
cat("Predictor:", predictor, "\n")
## Predictor: positive_impulse
cat("Pendiente:", round(coef(model)[2], 4), "\n")
## Pendiente: 0.0307
cat("R²:", round(summary(model)$r.squared, 4), "\n")
## R²: 0.4475
cat("p-valor:", round(summary(model)$coefficients[2, 4], 4), "\n")
## p-valor: 0
if(summary(model)$coefficients[2,4] < 0.05){
  cat("➡️ Relación significativa (p < 0.05)\n")
} else {
  cat("⚠️ Relación no significativa (p ≥ 0.05)\n")
}
## ➡️ Relación significativa (p < 0.05)
cat("\nInterpretación práctica:\n")
## 
## Interpretación práctica:
cat("Un aumento de una unidad en", predictor,
    "se asocia con un cambio promedio de",
    round(coef(model)[2], 2), "cm en la altura de salto (Jump Height).\n")
## Un aumento de una unidad en positive_impulse se asocia con un cambio promedio de 0.03 cm en la altura de salto (Jump Height).

11. Tabla resumen (GT)

broom::tidy(model) %>%
  select(term, estimate, std.error, statistic, p.value) %>%
  mutate(across(where(is.numeric), round, 3)) %>%
  gt() %>%
  gt::tab_header(
    title = "Resumen del Modelo Lineal Simple - CMJ",
    subtitle = paste("Variable predictora:", predictor)
  ) %>%
  gt::cols_label(
    term = "Término",
    estimate = "Coeficiente",
    std.error = "Error estándar",
    statistic = "Estadístico t",
    p.value = "p-valor"
  )
Resumen del Modelo Lineal Simple - CMJ
Variable predictora: positive_impulse
Término Coeficiente Error estándar Estadístico t p-valor
(Intercept) 19.865 1.107 17.952 0
positive_impulse 0.031 0.002 19.718 0

12. Conclusión

El modelo lineal simple permitió identificar la relación entre una variable biomecánica y la altura del salto CMJ.
Los resultados entregan información útil para comprender los factores de desempeño y orientar intervenciones en entrenamiento físico.