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dpikablog · 6 years ago

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Week 1 Assignment – Running an Analysis of variance (ANOVA)

Objective:

The assignment of the week deals with Analysis of variance. Given a dataset some form of Statistical Analysis test to be performed to check and evaluate its Statistical significance.

Before getting into the crux of the problem let us understand some of the important concepts

Hypothesis testing - It is one of the most important Inferential Statistics where the hypothesis is performed on the sample data from a larger population. Hypothesis testing is a statistical assumption taken by an analyst on the nature of the data and its reason for analysis. In other words, Statistical Hypothesis testing assesses evidence provided by data in favour of or against each hypothesis about the problem.

There are two types of Hypothesis

Null Hypothesis – The null hypothesis is assumed to be true until evidence indicate otherwise. The general assumptions made on the data (people with depression are more likely to smoke)

Alternate Hypothesis – Once stronger evidences are made, one can reject Null hypothesis and accept Alternate hypothesis (One needs to come with strong evidence to challenge the null hypothesis and draw proper conclusions).In this case one needs to show evidences such that there is no relation between people smoking and their depression levels

Example:

The Null hypothesis is that the number of cigarettes smoked by the person is dependent on the person’s depression level. Based on the p-value we make conclusions either to accept the null hypothesis or fail to accept the null hypothesis(accept alternative hypothesis)

Steps involved in Hypothesis testing:

1. Choose the Null hypothesis (H0 ) and alternate hypothesis (Ha)

2. Choose the sample

3. Assess the evidence

4. Draw the conclusions

The Null hypothesis is accepted/rejected based on the p-value significance level of test

If p<= 0.05, then reject the null hypothesis (accept the alternate hypothesis)

If p > 0.05 null hypothesis is accepted

Wrongly rejecting the null hypothesis leads to type one error

Sampling variability:

The measures of the sample (subset of population) varying from the measures of the population is called Sampling variability. In other words sample results changing from sample to sample.

Central Limit theorem:

As long as adequately large samples and an adequately large number of samples are used from a population,the distribution of the statistics of the sample will be normally distributed. In other words, the more the sample the accurate it is to the population parameters.

Choosing Statistical test:

Please find the below tabulation to identify what test can be done at a given condition. Some of the statistical tools used for this are

Chi-square test of independence, ANOVA- Analysis of variance, correlation coefficient.

Explanatory variables are input or independent variable And response variable is the output variable or dependent variable.

Explanatory Response Type of test

Categorical Categorical Chi-square test

Quantitative Quantitative Pearson correlation

Categorical Quantitative ANOVA

Quantitative Categorical Chi-square test

ANOVA:

Anova F test, helps to identify, Are the difference among the sample means due to true difference among the population or merely due to sampling variability.

F = variation among sample means / by variations within groups

Let’s implement our learning in python.

The Null hypothesis here is smoking and depression levels are unrelated

The Alternate hypothesis is smoking and depression levels are related.

# importing required libraries

import numpy as np

import pandas as pd

import statsmodels.formula.api as smf

data = pd.read_csv("my_data...nesarc.csv",low_memory=False)

#setting variables you will be working with to numeric

data['S3AQ3B1'] = data['S3AQ3B1'].convert_objects(convert_numeric=True)

#data['S3AQ3B1'] = pd.to_numeric(data.S3AQ3B1)

data['S3AQ3C1'] = data['S3AQ3C1'].convert_objects(convert_numeric=True)

#data['S3AQ3C1'] = pd.to_numeric(data.S3AQ3C1)

data['CHECK321'] = data['CHECK321'].convert_objects(convert_numeric=True)

#data['CHECK321'] = pd.to_numeric(data.CHECK321)

#subset data to young adults age 18 to 25 who have smoked in the past 12 months

sub1=data[(data['AGE']>=18) & (data['AGE']<=25) & (data['CHECK321']==1)]

#SETTING MISSING DATA

sub1['S3AQ3B1']=sub1['S3AQ3B1'].replace(9, np.nan)

sub1['S3AQ3C1']=sub1['S3AQ3C1'].replace(99, np.nan)

#recoding number of days smoked in the past month

recode1 = {1: 30, 2: 22, 3: 14, 4: 5, 5: 2.5, 6: 1}

sub1['USFREQMO']= sub1['S3AQ3B1'].map(recode1)

#converting new variable USFREQMMO to numeric

sub1['USFREQMO']= sub1['USFREQMO'].convert_objects(convert_numeric=True)

# Creating a secondary variable multiplying the days smoked/month and the number of cig/per day

sub1['NUMCIGMO_EST']=sub1['USFREQMO'] * sub1['S3AQ3C1']

sub1['NUMCIGMO_EST']= sub1['NUMCIGMO_EST'].convert_objects(convert_numeric=True)

ct1 = sub1.groupby('NUMCIGMO_EST').size()

print (ct1)

print(sub1['MAJORDEPLIFE'])

# using ols function for calculating the F-statistic and associated p value

The ols is the ordinary least square function takes the response variable NUMCIGMO_EST and its explanatory variable MAJORDEPLIFE, C is indicated to specify that it’s a categorical variable.

model1 = smf.ols(formula='NUMCIGMO_EST ~ C(MAJORDEPLIFE)', data=sub1)

results1 = model1.fit()

print (results1.summary())

Inference – Since the p value is greater than 0.05, we accept the null hypothesis that the sample means are statistically equal. And there is no association between depression levels and cigarette smoked.

sub2 = sub1[['NUMCIGMO_EST', 'MAJORDEPLIFE']].dropna()

print ('means for numcigmo_est by major depression status')

m1= sub2.groupby('MAJORDEPLIFE').mean()

print (m1)

print ('standard deviations for numcigmo_est by major depression status')

sd1 = sub2.groupby('MAJORDEPLIFE').std()

print (sd1)

Since the null hypothesis is true we conclude that the sample mean and standard deviation of the two samples are statistically equal. From the sample statistics we infer that the depression levels and smoking are unrelated to each other

Till now, we ran the ANOVA test for two levels of the categorical variable(MAJORDEPLIFE, 0,1) Lets now look at the categorical variable(ETHRACE2A,White, Black, American Indian, Asian native and Latino) having five levels and see how the differences in the mean are captured.

The problem here is, The F-statistic and ‘p’ value does not provide any insight as to why the null hypothesis is rejected when there are multiple levels in categorical Explanatory variable.

The significant ANOVA does not tell which groups are different from the others. To determine this we would need to perform post-hoc test for ANOVA

Why Post-hoc tests for ANOVA?

Post hoc test known as after analysis test, This is performed to prevent excessive Type one error. Not implementing this leads to family wise error rate, given by the formula

FWE = 1 – (1 – αIT)C

Where:

αIT = alpha level for an individual test (e.g. .05),

c = Number of comparisons.

Post-hoc tests are designed to evaluate the difference between the pair of means while protecting against inflation of type one error.

Let’s continue the code to perform post-hoc test.

# Importing the library

Import statsmodels.stats.multicomp as multi

# adding the variable of interest to a separate data frame

sub3 = sub1[['NUMCIGMO_EST', 'ETHRACE2A']].dropna()

# calling the ols function and passing the explanatory categorical and response variable

model2 = smf.ols(formula='NUMCIGMO_EST ~ C(ETHRACE2A)', data=sub3).fit()

print(model2.summary())

print ('means for numcigmo_est by major depression status')

m2= sub3.groupby('ETHRACE2A').mean()

print(m2)

print('standard deviations for numcigmo_est by major depression status')

sd2 = sub3.groupby('ETHRACE2A').std()

print(sd2)

# Include required parameters in the MultiComparison function and then run the post-hoc TukeyHSD test

mc1 = multi.MultiComparison(sub3['NUMCIGMO_EST'], sub3['ETHRACE2A'])

res1 = mc1.tukeyhsd()

print(res1.summary())

#statistics ANOVA DataAnalysis

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martingail11 · 5 years ago

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datasciencetechinfoplace · 3 years ago

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Hypothesis Testing & Anova Model

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manisa-group · 3 years ago

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expertlytics · 7 years ago

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———————————————————————— Analysis of Variance (ANOVA) is a statistical method used to test differences between two or more means. It may seem odd that the technique is called Analysis of Variance rather than Analysis of Means. — As you will see, the name is appropriate because inferences about means are made by analyzing variance. — A hypothesis test that enables us to test the equality of three or more means simultaneously using variance is called One way ANOVA. ... There is only one factor or independent variable in one way ANOVA whereas in the case of two-way ANOVA there are two independent variables. ———————————————————————— #data #bigdata #bigdataanalytics #dataanalysis #datascience #datascientist #datamining #dataprocessing #datavisualization #dataviz #machinelearning #anova #variance #algorithm #analysis #analytics #statistics #bayesianstatistics #studygram #learning #study #science #computers #computerscience #research #predictiveanalytics #математика #физика #наука #статистика (at United States)

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onlinespsscom · 5 years ago

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One-Way ANOVA

One-Way ANOVA ("analysis of variance") compares the means of two or more independent groups in order to determine whether there is statistical evidence that the associated population means are significantly different. One-Way ANOVA is a parametric test.This test is also known as:One-Factor ANOVAOne-Way Analysis of VarianceBetween Subjects ANOVAThe variables used in this test are known as:Dependent variableIndependent variable (also known as the grouping variable, or factor)This variable divides cases into two or more mutually exclusive levels, or groupsCommon UsesThe One-Way ANOVA is often used to analyze data from the following types of studies:Field studiesExperimentsQuasi-experimentsThe One-Way ANOVA is commonly used to test the following:Statistical differences among the means of two or more groupsStatistical differences among the means of two or more interventionsStatistical differences among the means of two or more change scoresNote: Both the One-Way ANOVA and the Independent Samples t-Test can compare the means for two groups. However, only the One-Way ANOVA can compare the means across three or more groups.Note: If the grouping variable has only two groups, then the results of a one-way ANOVA and the independent samples t-test will be equivalent. In fact, if you run both an independent samples t-test and a one-way ANOVA in this situation, you should be able to confirm that t2=F.Data RequirementsYour data must meet the following requirements:Dependent variable that is continuous (i.e., interval or ratio level)Independent variable that is categorical (i.e., two or more groups)Cases that have values on both the dependent and independent variablesIndependent samples/groups (i.e., independence of observations)There is no relationship between the subjects in each sample. This means that:subjects in the first group cannot also be in the second groupno subject in either group can influence subjects in the other groupno group can influence the other groupRandom sample of data from the populationNormal distribution (approximately) of the dependent variable for each group (i.e., for each level of the factor)Non-normal population distributions, especially those that are thick-tailed or heavily skewed, considerably reduce the power of the testAmong moderate or large samples, a violation of normality may yield fairly accurate p valuesHomogeneity of variances (i.e., variances approximately equal across groups)When this assumption is violated and the sample sizes differ among groups, the p-value for the overall F test is not trustworthy. These conditions warrant using alternative statistics that do not assume equal variances among populations, such as the Browne-Forsythe or Welch statistics (available via Options in the One-Way ANOVA dialog box).When this assumption is violated, regardless of whether the group sample sizes are fairly equal, the results may not be trustworthy for post hoc tests. When variances are unequal, post hoc tests that do not assume equal variances should be used (e.g., Dunnett’s C).No outliersNote: When the normality, homogeneity of variances, or outliers assumptions for One-Way ANOVA are not met, you may want to run the nonparametric Kruskal-Wallis test instead.Researchers often follow several rules of thumb for one-way ANOVA:Each group should have at least 6 subjects (ideally more; inferences for the population will be more tenuous with too few subjects)Balanced designs (i.e., the same number of subjects in each group) are ideal; extremely unbalanced designs increase the possibility that violating any of the requirements/assumptions will threaten the validity of the ANOVA F test.

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manisa-group · 3 years ago

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Cepat Murah, No.HP/WA: 0811-377-1086 Jasa Uji Validitas Corrected Item-total Correlation Cepat Dan Murah