One way anova python

How to Perform a One-Way Analysis of Variance (ANOVA) in Python

Analysis of Variance (ANOVA) is a statistical method used to test differences between two or more means. ANOVA compares the variance (or dispersion) between data samples to the variance within each particular sample itself. If the between-group variance is high and the within-group variance is low, it indicates that the means of different groups differ significantly.

One-way ANOVA is a type of ANOVA that studies the impact of a single factor on a response variable. If you have more than one independent variable, you would use a two-way ANOVA or N-way ANOVA.

In this tutorial, we’ll guide you through the process of performing a one-way ANOVA using Python. We’ll use the powerful statistical libraries SciPy and Statsmodels, both widely used for data analysis and manipulation.

Step 1: Import Required Libraries

We’ll need to import the necessary Python libraries.

import pandas as pd import numpy as np import matplotlib.pyplot as plt import seaborn as sns from scipy import stats import statsmodels.api as sm from statsmodels.formula.api import ols

Step 2: Loading Your Dataset

To perform a one-way ANOVA test, we will use an example dataset. The dataset is based on plant growth, where we have one independent variable (type of fertilizer) and one dependent variable (plant growth).

We’ll create this dataset using pandas:

data = df = pd.DataFrame(data)

Step 3: Exploratory Data Analysis

Before performing ANOVA, it’s always good to perform some exploratory data analysis (EDA) on the dataset. Let’s find the means for each group:

Visualize the data using box plots to understand the distribution of growth rates across different fertilizers:

Step 4: Check Assumptions of ANOVA

Before you can use ANOVA, there are several assumptions that the data must meet:

1. Normality: Each group of data should follow a normal distribution.
2. Homogeneity of variances: All groups must have the same variance.
3. Independence: Observations are independent of each other.

We won’t cover these assumptions in detail here, but it’s important to verify that your data meets them before proceeding with ANOVA.

Step 5: Calculate the One-Way ANOVA

We will now calculate the one-way ANOVA using two different methods.

Method 1: Using the scipy.stats Library

First, we’ll use the f_oneway function from the scipy.stats library:

F, p = stats.f_oneway(df['Fertilizer1'], df['Fertilizer2'], df['Fertilizer3']) print("F-value:", F) print("p-value:", p)

The f_oneway function returns two values: F-value and p-value. The F-value is a measure of how much the means of each group vary. The p-value is a measure of the probability that the differences in the means occurred by chance. If the p-value is less than 0.05, we can reject the null hypothesis that all groups have the same population mean.

Method 2: Using the statsmodels Library

The statsmodels library provides a more detailed output for one-way ANOVA. First, we need to reshape the data:

df_melt = pd.melt(df.reset_index(), id_vars=['index'], value_vars=['Fertilizer1', 'Fertilizer2', 'Fertilizer3']) df_melt.columns = ['index', 'treatments', 'value']

Next, we calculate the one-way ANOVA:

model = ols('value ~ C(treatments)', data=df_melt).fit() anova_table = sm.stats.anova_lm(model, typ=2) print(anova_table)

The ols function (Ordinary Least Squares) is used to fit the model, and the anova_lm function is used to calculate the ANOVA table.

Conclusion

Python provides powerful libraries that allow you to perform complex statistical tests, like one-way ANOVA, with just a few lines of code. It’s important to remember that while these tools are powerful, they’re not foolproof. Always ensure that your data meets the assumptions of the statistical test you’re using, and be cautious in your interpretation of the results.

Understanding and performing ANOVA is crucial in data analysis and decision making. It’s a versatile tool that helps us compare means of different groups and discover whether the differences are statistically significant.

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scipy.stats.f_oneway#

The one-way ANOVA tests the null hypothesis that two or more groups have the same population mean. The test is applied to samples from two or more groups, possibly with differing sizes.

Parameters : sample1, sample2, … array_like

The sample measurements for each group. There must be at least two arguments. If the arrays are multidimensional, then all the dimensions of the array must be the same except for axis.

axis int, optional

Axis of the input arrays along which the test is applied. Default is 0.

Returns : statistic float

The computed F statistic of the test.

pvalue float

The associated p-value from the F distribution.

Raised if all values within each of the input arrays are identical. In this case the F statistic is either infinite or isn’t defined, so np.inf or np.nan is returned.

Raised if the length of any input array is 0, or if all the input arrays have length 1. np.nan is returned for the F statistic and the p-value in these cases.

The ANOVA test has important assumptions that must be satisfied in order for the associated p-value to be valid.

1. The samples are independent.
2. Each sample is from a normally distributed population.
3. The population standard deviations of the groups are all equal. This property is known as homoscedasticity.

If these assumptions are not true for a given set of data, it may still be possible to use the Kruskal-Wallis H-test ( scipy.stats.kruskal ) or the Alexander-Govern test ( scipy.stats.alexandergovern ) although with some loss of power.

The length of each group must be at least one, and there must be at least one group with length greater than one. If these conditions are not satisfied, a warning is generated and ( np.nan , np.nan ) is returned.

If all values in each group are identical, and there exist at least two groups with different values, the function generates a warning and returns ( np.inf , 0).

If all values in all groups are the same, function generates a warning and returns ( np.nan , np.nan ).

The algorithm is from Heiman [2], pp.394-7.

R. Lowry, “Concepts and Applications of Inferential Statistics”, Chapter 14, 2014, http://vassarstats.net/textbook/

G.W. Heiman, “Understanding research methods and statistics: An integrated introduction for psychology”, Houghton, Mifflin and Company, 2001.

G.H. McDonald, “Handbook of Biological Statistics”, One-way ANOVA. http://www.biostathandbook.com/onewayanova.html

>>> import numpy as np >>> from scipy.stats import f_oneway 

Here are some data [3] on a shell measurement (the length of the anterior adductor muscle scar, standardized by dividing by length) in the mussel Mytilus trossulus from five locations: Tillamook, Oregon; Newport, Oregon; Petersburg, Alaska; Magadan, Russia; and Tvarminne, Finland, taken from a much larger data set used in McDonald et al. (1991).

>>> tillamook = [0.0571, 0.0813, 0.0831, 0.0976, 0.0817, 0.0859, 0.0735, . 0.0659, 0.0923, 0.0836] >>> newport = [0.0873, 0.0662, 0.0672, 0.0819, 0.0749, 0.0649, 0.0835, . 0.0725] >>> petersburg = [0.0974, 0.1352, 0.0817, 0.1016, 0.0968, 0.1064, 0.105] >>> magadan = [0.1033, 0.0915, 0.0781, 0.0685, 0.0677, 0.0697, 0.0764, . 0.0689] >>> tvarminne = [0.0703, 0.1026, 0.0956, 0.0973, 0.1039, 0.1045] >>> f_oneway(tillamook, newport, petersburg, magadan, tvarminne) F_onewayResult(statistic=7.121019471642447, pvalue=0.0002812242314534544) 

f_oneway accepts multidimensional input arrays. When the inputs are multidimensional and axis is not given, the test is performed along the first axis of the input arrays. For the following data, the test is performed three times, once for each column.

>>> a = np.array([[9.87, 9.03, 6.81], . [7.18, 8.35, 7.00], . [8.39, 7.58, 7.68], . [7.45, 6.33, 9.35], . [6.41, 7.10, 9.33], . [8.00, 8.24, 8.44]]) >>> b = np.array([[6.35, 7.30, 7.16], . [6.65, 6.68, 7.63], . [5.72, 7.73, 6.72], . [7.01, 9.19, 7.41], . [7.75, 7.87, 8.30], . [6.90, 7.97, 6.97]]) >>> c = np.array([[3.31, 8.77, 1.01], . [8.25, 3.24, 3.62], . [6.32, 8.81, 5.19], . [7.48, 8.83, 8.91], . [8.59, 6.01, 6.07], . [3.07, 9.72, 7.48]]) >>> F, p = f_oneway(a, b, c) >>> F array([1.75676344, 0.03701228, 3.76439349]) >>> p array([0.20630784, 0.96375203, 0.04733157]) 

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