week7
Saisree Mucharla
2024-10-15
Some questions which i found interesting after working on this dataset for 6 weeks are: - Which attributes are strongly correlated with obesity levels? - Do certain age groups or genders show higher obesity rates? - Can we identify groups of individuals who share common characteristics that are predictive of higher obesity levels?
Hypothesis 1
Is there a significant difference in the Body Mass Index (BMI) between males and females?
Step 1: Formulating the Hypotheses
- Null Hypothesis (H0): There is no significant difference in BMI between males and females.
- Mathematically: - The null hypothesis can be stated as:
\[ H_0: \mu_{\text{BMI, male}} = \mu_{\text{BMI, female}} \] - Alternative Hypothesis (H1): There is a significant difference in BMI between males and females. - Mathematically: - The alternative hypothesis can be stated as:
\[ H_1: \mu_{\text{BMI, male}} \neq \mu_{\text{BMI, female}} \]
Step 2: Choosing the Significance Level (α)
- The significance level (𝛼) represents the probability of having a Type I error, which is rejecting the null hypothesis when it is actually true. 𝛼=0.05
- This means that we are willing to accept a 5% risk of rejecting the null hypothesis when it is actually true. So, we accept that in 5 out of 100 samples, we might incorrectly conclude that there is a significant difference in BMI when there isn’t.
Step 3: Choosing the Power (1-β) and Effect Size
- Power refers to the probability of correctly rejecting the null hypothesis when it is false. - The higher the power, the better the test is at detecting a realeffect.
- Typical power: 0.80 (80%) — This means we want to have an 80% chance of detecting a true difference between males and females if it exists.
- Effect Size measures the magnitude of the difference between the two groups. For BMI, the effect size can be calculated as Cohen’s d, which is the difference between the two means divided by the pooled standard deviation.
- For this hypothesis, we might assume a medium effect size (d = 0.5)
Step 4: Sample Size Calculation - The sample size calculation ensures we have enough data to confidently perform the test. If we want a test with 80% power, an α of 0.05, and a medium effect size, we can calculate the necessary sample size using a statistical package like pwr in R.
library(pwr)
# Load the obesity dataset
# Load dataset
obesity <- read.csv("C:\\Users\\saisr\\Downloads\\statistics using R\\obesity.csv")
# View the first few rows of the dataset
head(obesity)## Gender Age Height Weight family_history_with_overweight FAVC FCVC NCP
## 1 Female 21 1.62 64.0 yes no 2 3
## 2 Female 21 1.52 56.0 yes no 3 3
## 3 Male 23 1.80 77.0 yes no 2 3
## 4 Male 27 1.80 87.0 no no 3 3
## 5 Male 22 1.78 89.8 no no 2 1
## 6 Male 29 1.62 53.0 no yes 2 3
## CAEC SMOKE CH2O SCC FAF TUE CALC MTRANS
## 1 Sometimes no 2 no 0 1 no Public_Transportation
## 2 Sometimes yes 3 yes 3 0 Sometimes Public_Transportation
## 3 Sometimes no 2 no 2 1 Frequently Public_Transportation
## 4 Sometimes no 2 no 2 0 Frequently Walking
## 5 Sometimes no 2 no 0 0 Sometimes Public_Transportation
## 6 Sometimes no 2 no 0 0 Sometimes Automobile
## NObeyesdad
## 1 Normal_Weight
## 2 Normal_Weight
## 3 Normal_Weight
## 4 Overweight_Level_I
## 5 Overweight_Level_II
## 6 Normal_Weight# Calculate sample size for a two-sample t-test
# Assumed effect size of 0.5 (medium effect), alpha of 0.05, and power of 0.80
effect_size <- 0.5
alpha_level <- 0.05
power <- 0.80
sample_size <- pwr.t.test(d = effect_size, sig.level = alpha_level, power = power, type = "two.sample")$n
sample_size## [1] 63.76561Step 5: Perform the Statistical Test (Two-Sample t-test) The t-test compares the means of two independent groups (males and females) to see if they are significantly different.
# Perform the two-sample t-test
library(dplyr)##
## Attaching package: 'dplyr'## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unionobesity <- obesity %>%
mutate(BMI = Weight / (Height^2))
head(obesity)## Gender Age Height Weight family_history_with_overweight FAVC FCVC NCP
## 1 Female 21 1.62 64.0 yes no 2 3
## 2 Female 21 1.52 56.0 yes no 3 3
## 3 Male 23 1.80 77.0 yes no 2 3
## 4 Male 27 1.80 87.0 no no 3 3
## 5 Male 22 1.78 89.8 no no 2 1
## 6 Male 29 1.62 53.0 no yes 2 3
## CAEC SMOKE CH2O SCC FAF TUE CALC MTRANS
## 1 Sometimes no 2 no 0 1 no Public_Transportation
## 2 Sometimes yes 3 yes 3 0 Sometimes Public_Transportation
## 3 Sometimes no 2 no 2 1 Frequently Public_Transportation
## 4 Sometimes no 2 no 2 0 Frequently Walking
## 5 Sometimes no 2 no 0 0 Sometimes Public_Transportation
## 6 Sometimes no 2 no 0 0 Sometimes Automobile
## NObeyesdad BMI
## 1 Normal_Weight 24.38653
## 2 Normal_Weight 24.23823
## 3 Normal_Weight 23.76543
## 4 Overweight_Level_I 26.85185
## 5 Overweight_Level_II 28.34238
## 6 Normal_Weight 20.19509t_test_result <- t.test(BMI ~ Gender, data = obesity , var.equal = TRUE)
# View the results of the t-test
t_test_result##
## Two Sample t-test
##
## data: BMI by Gender
## t = 2.439, df = 2109, p-value = 0.01481
## alternative hypothesis: true difference in means between group Female and group Male is not equal to 0
## 95 percent confidence interval:
## 0.1664841 1.5327651
## sample estimates:
## mean in group Female mean in group Male
## 30.13000 29.28038- Interpreting the results: There is a statistically significant difference in BMI between females and males (p < 0.05). Specifically, females have a higher mean BMI (30.13) compared to males (29.28). The confidence interval indicates that this difference is likely between approximately 0.17 and 1.53 BMI units, reinforcing the finding that gender appears to influence BMI in your dataset.
Visualization for hypotheses 1
# Load necessary libraries
library(dplyr)
library(ggplot2)
# Create a summary dataset for means and confidence intervals
summary_data <- obesity %>%
group_by(Gender) %>%
summarise(
mean_BMI = mean(BMI),
sd_BMI = sd(BMI),
n = n(),
se_BMI = sd_BMI / sqrt(n), # Standard Error
ci_lower = mean_BMI - qt(0.975, df = n - 1) * se_BMI, # Lower CI
ci_upper = mean_BMI + qt(0.975, df = n - 1) * se_BMI # Upper CI
)
# Create the bar plot with error bars
ggplot(summary_data, aes(x = Gender, y = mean_BMI, fill = Gender)) +
geom_bar(stat = "identity", position = position_dodge(), width = 0.6) + # Create the bar plot
geom_errorbar(aes(ymin = ci_lower, ymax = ci_upper), width = 0.2, position = position_dodge(0.6), color = "black") + # Add error bars
labs(title = "Mean BMI Comparison between Males and Females",
x = "Gender",
y = "Mean BMI") +
theme_minimal() +
scale_fill_manual(values = c("lightblue", "lightpink")) + # Customize colors
theme(legend.position = "none") # Re
Insights
- The bar plot clearly shows the average BMI for both males and females. From the data provided earlier, it appears that: Mean BMI for Males: Approximately 29.28 Mean BMI for Females: Approximately 30.13 This indicates that, on average, females have a higher BMI compared to males in your dataset.
- The error bars represent the 95% confidence intervals for the mean BMI of each gender. If the error bars do not overlap, this suggests a statistically significant difference between the groups at the 0.05 significance level. In this case, since the confidence intervals for males and females do not overlap (lower bound of females’ CI is above the upper bound of males’ CI), we can conclude that the difference in mean BMI is statistically significant.
Conclusion for Hypothesis 1
- After performing the two-sample t-test and interpreting the results: You either reject or fail to reject the null hypothesis based on the p-value. You interpret these results in the context of the research question, explaining the potential implications of the difference in BMI between males and females in this dataset.
Hypothesis 2
Does the level of physical activity significantly influence obesity levels?
Step 1: Formulating the Hypotheses
- Null Hypothesis (H0): Physical activity level and obesity status are independent of each other (i.e., physical activity has no effect on obesity).
- mathematically: The null hypothesis can be stated as:
\[ H_0: \text{Physical Activity} \perp \text{Obesity Status} \] - Alternative Hypothesis (H1): Physical activity level and obesity status are not independent (i.e., physical activity has an effect on obesity). - Mathematically: The alternative hypothesis can be stated as:
\[ H_1: \text{Physical Activity} \not\perp \text{Obesity Status} \]
Step 2: Choosing the Significance Level (α) - For Fisher’s significance testing, a significance level (α) of 0.05 is typically chosen. - This means that we are willing to accept a 5% probability of incorrectly rejecting the null hypothesis (a Type I error). - If the p-value from our test is less than 0.05, we will reject the null hypothesis and conclude that physical activity levels and obesity are not independent.
Step 3: Choosing the Statistical Test - Since both physical activity level and obesity status are categorical variables, a Chi-square test of independence is appropriate. Physical Activity Level could be divided into categories (e.g., low, medium, high). Obesity Level could also be categorized (e.g., normal, overweight, obese).
Step 4: Creating a Contingency Table - creating a contingency table that shows the counts of individuals for each combination of physical activity and obesity level.
# Create a contingency table
table_physical_activity <- table(obesity$FAF, obesity$NObeyesdad)
# View the contingency table
table_physical_activity##
## Insufficient_Weight Normal_Weight Obesity_Type_I Obesity_Type_II
## 0 34 80 81 28
## 9.6e-05 0 0 0 0
## 0.000272 0 0 0 0
## 0.000454 0 0 0 0
## 0.001015 0 0 0 0
## 0.001086 0 0 1 0
## 0.001272 0 0 0 0
## 0.001297 0 0 0 0
## 0.00203 0 0 1 0
## 0.00342 0 0 0 0
## 0.003938 0 0 1 0
## 0.005405 0 0 1 0
## 0.005939 0 0 0 0
## 0.006265 0 0 0 0
## 0.00705 0 0 0 0
## 0.008013 0 0 0 0
## 0.008127 0 0 0 0
## 0.008368 0 0 1 0
## 0.010183 0 0 0 0
## 0.011161 0 0 0 0
## 0.011477 0 0 1 0
## 0.011519 0 0 1 0
## 0.01437 0 0 0 0
## 0.01586 0 0 0 0
## 0.01682 0 0 0 0
## 0.017804 0 0 0 0
## 0.019404 0 0 0 0
## 0.02006 0 0 0 0
## 0.02112 0 0 0 0
## 0.022598 0 0 1 0
## 0.022958 0 0 0 0
## 0.023574 1 0 0 0
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## 0.026033 0 0 0 0
## 0.026142 0 0 1 0
## 0.027101 0 0 0 0
## 0.027433 1 0 0 0
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## 0.029603 0 0 0 1
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## 0.054238 0 0 0 1
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## 2.870127 1 0 0 0
## 2.876696 0 0 0 0
## 2.877473 0 0 0 0
## 2.878414 0 0 0 0
## 2.89118 0 0 1 0
## 2.891986 0 0 1 0
## 2.892922 0 0 0 0
## 2.931527 0 0 0 0
## 2.936551 0 0 0 0
## 2.939733 0 0 0 0
## 2.971832 0 0 0 0
## 2.998981 0 0 1 0
## 2.999918 0 0 1 0
## 3 4 41 13 0
##
## Obesity_Type_III Overweight_Level_I Overweight_Level_II
## 0 87 40 61
## 9.6e-05 1 0 0
## 0.000272 1 0 0
## 0.000454 1 0 0
## 0.001015 1 0 0
## 0.001086 0 0 0
## 0.001272 1 0 0
## 0.001297 1 0 0
## 0.00203 0 0 0
## 0.00342 0 1 0
## 0.003938 0 0 0
## 0.005405 0 0 0
## 0.005939 1 0 0
## 0.006265 1 0 0
## 0.00705 0 1 1
## 0.008013 1 0 0
## 0.008127 1 0 0
## 0.008368 0 0 0
## 0.010183 1 0 0
## 0.011161 1 0 0
## 0.011477 0 0 0
## 0.011519 0 0 0
## 0.01437 1 0 0
## 0.01586 2 0 0
## 0.01682 1 0 0
## 0.017804 1 0 0
## 0.019404 1 0 0
## 0.02006 1 0 0
## 0.02112 0 0 1
## 0.022598 0 0 0
## 0.022958 0 0 1
## 0.023574 0 0 0
## 0.025787 1 0 0
## 0.026033 2 0 0
## 0.026142 0 0 0
## 0.027101 1 0 0
## 0.027433 0 0 0
## 0.028202 0 0 0
## 0.029603 0 0 0
## 0.029728 2 0 0
## 0.030541 1 0 0
## 0.033328 0 0 0
## 0.033745 0 0 1
## 0.03465 1 0 0
## 0.035091 0 0 0
## 0.035928 1 0 0
## 0.038809 0 0 1
## 0.039207 0 0 1
## 0.043101 1 0 0
## 0.043412 0 0 0
## 0.043689 1 0 0
## 0.045246 1 0 0
## 0.04525 1 0 0
## 0.045651 0 1 0
## 0.046836 0 1 0
## 0.047738 0 0 1
## 0.05093 1 0 0
## 0.051097 1 0 0
## 0.054238 0 0 0
## 0.057758 0 0 0
## 0.061366 0 1 0
## 0.06275 0 0 0
## 0.062932 0 0 0
## 0.063383 1 0 0
## 0.065264 0 0 0
## 0.06582 1 0 0
## 0.067329 1 0 0
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## 0.095517 0 1 0
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## 0.098043 1 0 0
## 0.10032 1 0 0
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## 0.10297 0 0 0
## 0.104451 0 1 0
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## 0.107078 0 0 0
## 0.107981 0 0 0
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## 0.108948 0 0 0
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## 0.110174 0 1 1
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## 0.120165 0 0 0
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## 0.131371 0 0 0
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## 0.143955 2 0 0
## 0.144627 0 0 0
## 0.14495 0 0 0
## 0.145687 0 0 0
## 0.146919 0 0 1
## 0.148628 0 1 0
## 0.15136 0 1 0
## 0.152713 1 0 0
## 0.155579 0 0 1
## 0.159255 1 0 0
## 0.162083 1 0 0
## 0.162279 1 0 0
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## 2 1 22 10
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## 2.892922 0 1 1
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## 2.998981 0 0 0
## 2.999918 0 0 0
## 3 0 10 7Step 5: Performing the Chi-Square Test
# Perform the Chi-square test
chi_sq_test <- chisq.test(table_physical_activity)## Warning in chisq.test(table_physical_activity): Chi-squared approximation may
## be incorrect# View the test results
chi_sq_test##
## Pearson's Chi-squared test
##
## data: table_physical_activity
## X-squared = 7678.3, df = 7134, p-value = 4.275e-06- Interpreting the results:
- X-squared = 7678.3: This value represents the test statistic calculated from your data. A higher Chi-squared statistic indicates a greater difference between the observed frequencies (actual data) and the expected frequencies (what we would expect under the null hypothesis of no association). This suggests a significant deviation from the null hypothesis.
- df = 7134: This indicates the degrees of freedom associated with your Chi-squared test.
- p-value = 4.275e-06: This extremely small p-value indicates a statistically significant association between the variables in your contingency table. When the p-value is much less than the common alpha level of 0.05, it suggests that you should reject the null hypothesis.
Visualization for hypotheses 2
# Assuming you have the necessary libraries
library(ggplot2)
library(dplyr)
# Sample data for demonstration (replace with your actual contingency table)
# Create a sample contingency table for physical activity levels and gender
set.seed(123) # For reproducibility
Gender <- sample(c("Male", "Female"), 1000, replace = TRUE)
Activity_Level <- sample(c("Low", "Moderate", "High"), 1000, replace = TRUE)
table_physical_activity <- table(Gender, Activity_Level)
# Create a data frame from the contingency table
df_activity <- as.data.frame(table_physical_activity)
# Create a simple bar plot
ggplot(df_activity, aes(x = Activity_Level, fill = Gender)) +
geom_bar(stat = "count") +
labs(title = "Count of Physical Activity Levels by Gender",
x = "Physical Activity Level",
y = "Count",
fill = "Gender") +
theme_minimal() +
theme(axis.text.x = element_text(angle = 45, hjust = 1))
Insights
- If Males have significantly more participants in the Moderate or High activity levels compared to Females, this could suggest a trend in physical activity behavior. Conversely, if Females show higher counts in Low activity levels, this could indicate that a higher proportion of females are less active compared to males
- If one gender consistently appears more active than the other across all levels, this might raise questions regarding cultural, societal, or individual factors that influence physical activity.
Conclusion for Hypothesis 2
Conclusion (if p-value < 0.05): There is significant evidence to suggest that physical activity levels influence obesity. Individuals with higher physical activity levels may have lower probabilities of being classified as obese.
Conclusion (if p-value ≥ 0.05): There is no significant evidence to suggest that physical activity levels are associated with obesity. In this case, other factors may play a more critical role in influencing obesity status.