One Way ANOVA

Dr.Shovan padhy, MBBS, MDDM 1st yr (Senior Resident)

NIMS, Hyderabad

Overview• Introduction.• Why ANOVA instead of multiple t-tests?• One way ANOVA.• Assumptions of One way ANOVA.• Steps in One way ANOVA.• Example.• Conclusion.

Introduction• ANOVA is an abbreviation for the full name of the method:

Analysis Of Variance.

• Invented by R.A. Fisher in the 1918.

• ANOVA is used to test the significance of the difference between more than two sample means.

• Name “ANOVA” is a misnomer as it compares mean to check variance between group.

Summary Table of Statistical tests

Level of Measurement

Sample CharacteristicsCorrelation

1 Sample2 Sample K Sample (i.e., >2)

Independent Dependent Independent Dependent

Categorical or Nominal

Χ2 or bi-nomina

l

Χ2 Macnarmar’s Χ2

Χ2 Cochran’s Q  

Rank or Ordinal

  Mann Whitney U

Wilcoxin Matched

Pairs Signed Ranks

Kruskal Wallis H

Friedman’s ANOVA

Spearman’s rho

Parametric (Interval &

Ratio)

z test or

t test

t test between groups

t test within groups

1 way ANOVA between groups

1 way ANOVA (within or repeated measure)

Pearson’s r

Factorial (2 way) ANOVA   

Χ2

Why ANOVA instead of multiple t-tests?• If you are comparing means between more than two groups,

we can choose two sample t-tests to compare the mean of one group with the mean of the other groups?:Before ANOVA, this was the only option available to compare means between more than two groups.

• The problem with the multiple t-tests approach is that as the number of groups increases, the number of two sample t-tests also increases.

• As the number of tests increases the probability of making a Type I error also increases.

One way ANOVA• One way ANOVA (=F test) compares the mean of two or more

group whenever there is one independent variable is involved.• It finds out whether there are any statistical significance

difference between their group means.• If more then one independent variable is involved then it is

called as N way ANOVA.• One way ANOVA specifically tests the null hypothesis.• H0 = u1 = u2 = u3 = uk , u = group mean & k= no. of groups.• If One way ANOVA shows a statistical significant result it

means HA is true.

Variables In One way ANOVA

• In an ANOVA, there are two kinds of variables: independent and dependent

• The independent variable is controlled or manipulated by the researcher.

• It is a categorical (discrete) variable used to form the groupings of observations.

• There are two types of independent variables: active and attribute.

• If the independent variable is an active variable then we manipulate the values of the variable to study its affect on another variable. • For example, anxiety level is an active independent variable.

• An attribute independent variable is a variable where we do not alter the variable during the study. • For example, we might want to study the effect of age on

weight. We cannot change a person’s age, but we can study people of different ages and weights.

• In the One-way ANOVA, only one independent variable is considered, but there are two or more (theoretically any finite number) levels of the independent variable.

• The independent variable is typically a categorical variable.

• The independent variable (or factor) divides individuals into two or more groups or levels.

• The procedure is a One-way ANOVA, since there is only one independent variable.

• The (continuous) dependent variable is defined as the variable that is, or is presumed to be, the result of manipulating the independent variable.

• In the One-way ANOVA, there is only one dependent variable – and hypotheses are formulated about the means of the groups on that dependent variable.

• The dependent variable differentiates individuals on quantitative (continuous) dimension.

Assumptions of One way ANOVA1) All populations involved follow a normal distribution.

2) Homogeneity of variances: The variance within each group should be equal for all groups.

3) Independence of error: The error (variation of each value around its own group mean) should be independent for each value.

4) Only ONE independent variable should be checked whether it produces a significant difference between the groups.

Example of One way ANOVA

Group A Group B Group C160,110,118,124,132

122,136,124,126,120,138

148,126,124,128,140

N1= 5 N2= 6 N3= 5Mean=128.8 Mean= 127.66 Mean=133.2

EXAMPLE: A study conducted to assess & compare the effect of Treatment A vs Treatment B vs Treatment C on SBP in a specified population.

ANOVA

One way ANOVA

Three way ANOVA

Effect of Drugs on SBP

Two way ANOVA

Effect of Diet & Drugs on SBP

Effect of Exercise, Drugs, Diet on SBP

Steps in One way ANOVA

2. State Alpha3. Calculate degrees of Freedom

4. Calculate test statistic- Calculate variance between samples- Calculate variance within the samples- Calculate F statistic

1. State null & alternative hypotheses

Example- one way ANOVA

Example: A investigator wants to find out the analgesic effect of aspirin vs diclofenac vs ibuprofen in a group of population with equal variances.

Aspirin Diclofenac Ibuprofen1 5 94 10 37 2 29 1 43 7 2

Steps Involved

1.Null hypothesis – No significant difference in the means of 3 samples

2. State Alpha i.e 0.05

3. Calculate degrees of Freedom k-1 & n-k = 2 & 12

4. State decision rule Table value of F at 5% level of significance for d.f 2 & 12 is 3.88The calculated value of F > 3.88 , H0 will be rejected

5. Calculate test statistic

One way ANOVA: Table

Source of Variation

SS (Sum of Squares)

Degrees of Freedom

MS (Mean Square)

Variance Ratio of F

Between Samples

SSB k-1 MSB= SSB/(k-1)

MSB/MSW

Within Samples

SSW n-k MSW= SSW/(n-k)

Total SS(Total) n-1

Calculating variance BETWEEN samples

1. Calculate the mean of each sample.2. Calculate the Grand mean.3. Take the difference between means of various samples &

grand average.4. Square these deviations & obtain total which will give sum

of squares between samples (SSC)5. Divide the total obtained in step 4 by the degrees of freedom

to calculate the mean sum of square between samples (MSC).

Aspirin Diclofenac Ibuprofen1 7 24 5 97 10 39 2 23 1 4

Total 24M1= 4.8

25M2 = 5

20M3 = 4

4.8+ 5+ 4 3

Grand average = = 4.6

Variance BETWEEN samples (M1=4.8, M2=5,M3=4)

Sum of squares between samples (SSC) = n1 (M1 – Grand avg)2 + n2 (M2– Grand avg)2 + n3(M3– Grand avg)2

5 ( 4.8 - 4.6) 2 + 5 ( 5 - 4.8) 2 + 5 ( 4.6 - 4.8) 2 = 0.6

Calculation of Mean sum of squares between samples (MSB) =0.6/2 = 0.3

k= No of Samples, n= Total No of observations

Calculating Variance WITHIN Samples

1. Calculate mean value of each sample.2. Take the deviations of the various items in a sample from the

mean values of the respective samples.3. Square these deviations & obtain total which gives the sum

of square within the samples (SSE) 4. Divide the total obtained in 3rd step by the degrees of

freedom to calculate the mean sum of squares within samples (MSE).

Variance WITHIN samples (M1= 4.8, M2= 5,M3= 4)

X1 (X1 – M1)2 X2 (X2– M2)2 X3 (X3– M3)2

1 14.4 7 4 2 44 0.64 5 0 9 257 4.84 10 25 3 19 17.64 2 9 2 43 3.24 1 16 4 0

40.76 54 34

Sum of squares within samples (SSE) = 40.76 + 54 +34 =128.76 Calculation of Mean Sum Of Squares within samples (MSW) = 128.76/12 = 10.73

The mean sum of squares

1kSSCMSC

knSSEMSE

Calculation of MSC-Mean sum of Squares between samples

Calculation of MSEMean Sum Of Squares within samples

k= No of Samples, n= Total No of observations

Calculation of F statsitics

groupswithinyVariabilitgroupsbetweenyVariabilitF

Compare the F-statistic value with F(critical) value which is obtained by looking for it in F distribution tables against degrees of freedom. The calculated value of F > table valueH0 is rejected

• F Value = MSB/MSW = 0.3/10.73 = 0.02.

The Table value of F at 5% level of significance for d.f 2 & 12 is 3.88The calculated value of F < table valueH0 is accepted. Hence there no is significant difference in sample means

Within-Group Variance

Between-Group Variance

Between-group variance is large relative to the within-group variance, so F statistic will be larger & > critical value, therefore statistically significant . Conclusion – At least one of group means is significantly different from other group means

Within-Group Variance

Between-Group Variance

Within-group variance is larger, and the between-group variance smaller, so F will be smaller (reflecting the likely-hood of no significant differences between these 3 sample means)

Post-hoc Tests

• Used to determine which mean or group of means is/are significantly different from the others (significant F)

• Depending upon research design & research question: Bonferroni (more powerful) Only some pairs of sample means are to be testedDesired alpha level is divided by no. of comparisons

Tukey’s HSD Procedure when all pairs of sample means are to be tested

Scheffe’s Procedure (when sample sizes are unequal)

Application of ANOVA

• ANOVA is designed to detect differences among means from populations subject to different treatments.

• ANOVA is a joint test, the equality of several population means is tested simultaneously or jointly.

• ANOVA tests for the equality of several population means by looking at two estimators of the population variance (hence, analysis of variance).

Conclusion• The one-way analysis of variance is used where there is a

single factor that will be set to three or more levels.• t is not appropriate to analyse such data by repeated t-tests as

this will raise the risk of false positives above the acceptable level of 5 per cent.

• If the ANOVA produces a significant result, this only tells us that at least one level produces a different result from one of the others.

• Follow-up tests needs to be carried out to find out which group differs from each other.

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