ANCOVA is an extension of ANOVA (Analysis of Variance) that combines blocks of regression analysis and ANOVA. Which makes it Analysis of Covariance.
ANCOVA is an extension of ANOVA (Analysis of Variance) that combines blocks of regression analysis and ANOVA. Which makes it Analysis of Covariance.
e.g. Think of ANCOVA as a way to level the playing field. It adjusts for other factors (covariates) so we can clearly see the impact of our main variable on the outcome.
e.g. Think of ANCOVA as a way to compare groups while keeping other factors constant. It’s like comparing the performance of students from different schools while accounting for their study hours.
In scientific research, a hypothesis is crucial. It guides the investigation
and helps focus on a specific research question. A hypothesis gives the study a clear direction
, making sure that data collection and analysis are purposeful and meaningful. Without a hypothesis, research may lack direction and results may be hard to interpret.
e.g Imagine you’re guessing the number of candies in a jar. Hypothesis testing is like checking if your guess is close enough to the actual number or if it’s way off.
ANOVA (Analysis of Variance), MANOVA (Multivariate Analysis of Variance)
, and ANCOVA (Analysis of Covariance)
are statistical techniques used to analyze the relationship between variables. ANOVA
compares means between groups, MANOVA
extends this to multiple dependent variables, and ANCOVA
accounts for the effect of additional variables (covariates)
on the relationship. These techniques help researchers understand the significance of differences between groups and the impact of covariates on the outcomes.
ANCOVA is particularly useful in situations where:
2. A medical study is investigating the effectiveness of three different treatments for reducing cholesterol levels. The researchers also want to account for the potential influence of patients’ age and baseline cholesterol levels.
Imagine you’re a doctor studying the effect of a new drug. ANCOVA helps you see the drug’s impact while considering patients’ ages. Similarly, in education, it can show how different teaching methods work while accounting for students’ prior knowledge.
Cholesterol Reduction
~ Treatment Type
+ Age
+ Baseline Cholesterol
The previous examples show how to study the connection between a main result and various influencing factors while considering the effects of other related variables.
ANCOVA has been widely used in various fields, including:
different treatments
while controlling for patient characteristics
different teaching methods
on student outcomes
while accounting for student IQ
statsmodels
library to see how teaching methods affect scores, considering study hours.customer behaviour
and marketing strategies
while controlling for demographic variables
This code snippet imports necessary libraries reads a CSV file ‘teengamb.csv’ into a pandas DataFrame, and prints the first few rows of the dataset.
import pandas as pd
from statsmodels.formula.api import ols
import statsmodels.api as sm
teengamb = pd.read_csv('teengamb.csv')
# View the first few rows of the dataset
print(teengamb.head())
index | sex | status | income | verbal | gamble |
---|---|---|---|---|---|
0 | 1 | 51 | 2.0 | 8 | 0.0 |
1 | 1 | 28 | 2.5 | 8 | 0.0 |
2 | 1 | 37 | 2.0 | 6 | 0.0 |
3 | 1 | 28 | 7.0 | 4 | 7.3 |
4 | 1 | 65 | 2.0 | 8 | 19.6 |
Teenage Gambling
)The teengamb
dataset from R (in the faraway
package) contains information related to teenage gambling in Britain. Here are the columns typically found in this dataset:
7.3 euros
per year1£ pound= 108.70 ₹
0 = male, 1 = female
. 0 = low and 100 = high
est. 1= lowest to 10= highest
Each column provides specific insights into factors that may influence teenage gambling behaviour, facilitating various statistical analyses and research studies.
This code specifies and fits an ANCOVA (Analysis of Covariance) model using the ols
function from statsmodels.formula.api
. It examines the relationship between the dependent variable ‘gamble’ (teenage gambling expenditure) and several independent variables ('income', 'sex', 'status', 'verbal')
. The .fit()
the method fits the model to the data, and .summary()
provides a detailed summary of the model statistics, including coefficients, standard errors, t-statistics, p-values, and confidence intervals.
The formula:
Variable: ~ Group Variable + Covariate
describes the structure of an Analysis of Covariance (ANCOVA) model. Here’s a detailed explanation:
The purpose of ANCOVA is to adjust the dependent variable for the influence of the covariate, isolating the effect of the group variable. Here’s how it works:
‘Dependent Variable’ = gamble
‘Group Variable’ = ‘sex’,
‘Covariates‘ (Independent Variables)= income
, sex
, status
, and verbal
# Specify the ANCOVA model
model = ols('gamble ~ income + sex + status + verbal', data=teengamb).fit()
#summarise model
mode.summary()
Dep. Variable: | gamble | R-squared: | 0.527 |
---|---|---|---|
Model: | OLS | Adj. R-squared: | 0.482 |
Method: | Least Squares | F-statistic: | 11.69 |
Date: | Mon, 15 Jul 2024 | Prob (F-statistic): | 1.81e-06 |
Time: | 14:52:21 | Log-Likelihood: | -210.78 |
No. Observations: | 47 | AIC: | 431.6 |
Df Residuals: | 42 | BIC: | 440.8 |
Df Model: | 4 | ||
Covariance Type: | nonrobust |
coef | std err | t | P>|t| | [0.025 | 0.975] | |
---|---|---|---|---|---|---|
Intercept | 22.5557 | 17.197 | 1.312 | 0.197 | -12.149 | 57.260 |
income | 4.9620 | 1.025 | 4.839 | 0.000 | 2.893 | 7.031 |
sex | -22.1183 | 8.211 | -2.694 | 0.010 | -38.689 | -5.548 |
status | 0.0522 | 0.281 | 0.186 | 0.853 | -0.515 | 0.620 |
verbal | -2.9595 | 2.172 | -1.362 | 0.180 | -7.343 | 1.424 |
Omnibus: | 31.143 | Durbin-Watson: | 2.214 |
---|---|---|---|
Prob(Omnibus): | 0.000 | Jarque-Bera (JB): | 101.046 |
Skew: | 1.604 | Prob(JB): | 1.14e-22 |
Kurtosis: | 9.427 | Cond. No. | 264. |
sum_sq | df | F | PR(>F) | |
---|---|---|---|---|
sex | 3735.790512 | 1.0 | 7.256053 | 0.010112 |
income | 12056.238564 | 1.0 | 23.416920 | 0.000018 |
status | 17.775781 | 1.0 | 0.034526 | 0.853487 |
verbal | 955.734110 | 1.0 | 1.856329 | 0.180311 |
Residual | 21623.767055 | 42.0 | NaN | NaN |
Think of the output metrics like a report card. The ‘sum of squares’ shows the total variation, ‘degrees of freedom’ indicate the number of comparisons made, ‘F-value’ tells you how strong the effect is, and ‘p-value’ shows if the results are significant, like a passing grade.
sum_sq | df | F | PR(>F) | |
---|---|---|---|---|
sex | 3735.790512 | 1.0 | 7.256053 | 0.010112 |
income | 12056.238564 | 1.0 | 23.416920 | 0.000018 |
status | 17.775781 | 1.0 | 0.034526 | 0.853487 |
verbal | 955.734110 | 1.0 | 1.856329 | 0.180311 |
Residual | 21623.767055 | 42.0 | NaN | NaN |
The ANCOVA analysis of the teengamb dataset reveals that both gender and income significantly influence gambling expenditure among teenagers in Britain, while socioeconomic status and verbal IQ do not show significant effects.
ANCOVA is an extension of ANOVA (Analysis of Variance) that combines blocks of regression analysis and ANOVA. Which makes it Analysis of Covariance.
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