How to understand the Analysis of variance?
Analysis of variance is a statistical technique by which the significance of the difference between more than two sample means is tested. With the T-Test and Z-Test, the significance of the difference between two samples' mean is tested but when the test is applied to more than two samples these methods are not appropriate and the ANOVA technique is used. Analysis of variance was developed by famous statistician R.A . Fisher in 1932. It is used widely in natural, social, and agricultural investigations. According to Sir R.A. Fisher, the Analysis of variance is the separation of the variance ascribed to one group of causes from the variance ascribed to another group. Analysis of Variance (ANOVA) is a statistical technique used to analyze and compare the means of two or more groups to determine if there are statistically significant differences among them. ANOVA is particularly useful when you want to test whether variations between group means are due to actual differences or if they could have occurred by random chance alone. ANOVA can be thought of as an extension of the t-test for comparing the means of two groups to situations involving multiple groups.
Here are the key components and concepts of ANOVA:
Groups or Treatments: ANOVA involves comparing the means of two or more groups or treatments. These groups can represent different categories, levels, or experimental conditions. For example, in a drug efficacy study, groups could have different drug dosage levels.
Null Hypothesis (H0): The null hypothesis in ANOVA assumes that there are no statistically significant differences among the group means. In other words, all groups have the same population mean.
Alternative Hypothesis (H1): The alternative hypothesis, which is complementary to the null hypothesis, suggests that at least one group mean is statistically different from the others.
Variability: ANOVA takes into account two types of variability: within-group variability and between-group variability. Within-group variability represents the variation of data points within each group, while between-group variability represents the variation of group means from one another.
F-Statistic: ANOVA uses the F-statistic to test whether the differences between group means are statistically significant. The F-statistic is calculated by dividing the between-group variability by the within-group variability. If the F-statistic is sufficiently large, it suggests that at least one group is different from the others.
Degrees of Freedom: ANOVA involves two degrees of freedom parameters: degrees of freedom between groups (DF between) and degrees of freedom within groups (DF within). These are used in the calculation of the F-statistic.
P-Value: The F-statistic is used to calculate a p-value, which indicates the probability of obtaining the observed differences between group means (or more extreme differences) if the null hypothesis were true. A small p-value (typically less than 0.05) suggests that the differences are statistically significant, leading to the rejection of the null hypothesis.
Post-Hoc Tests: If ANOVA results in a rejection of the null hypothesis (indicating differences among groups), post-hoc tests can be conducted to determine which specific group means are different from each other. Common post-hoc tests include Tukey's Honestly Significant Difference (HSD) and Bonferroni correction.
Assumptions: ANOVA assumes that the data in each group are normally distributed and that the variances among the groups are approximately equal (homoscedasticity). Violations of these assumptions can impact the validity of ANOVA results.
Applications: ANOVA is widely used in various fields, including experimental psychology, biology, medicine, economics, and social sciences, to compare means across multiple groups in a systematic and statistically rigorous manner.
In summary, Analysis of Variance (ANOVA) is a statistical method for comparing the means of multiple groups to assess whether there are significant differences among them. It helps researchers determine whether observed variations are due to real effects or could have occurred by chance. ANOVA is a valuable tool for hypothesis testing and drawing conclusions in experimental and observational studies involving multiple groups or treatments.
Assumption of Analysis of Variance
Techniques of analysis of variance
ANOVA in Python
data = {‘ac’ : [70, 67, 65, 75, 76, 73, 69, 68, 70, 76, 77, 75, 85,
86, 85, 76, 75, 73, 95, 94, 89, 94, 93, 91],
‘teach’ : [1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4,
4, 4, 4, 4, 4]}
df = pd.DataFrame(data)
df
df.info()
<class ‘pandas.core.frame.DataFrame’>
RangeIndex: 24 entries, 0 to 23
Data columns (total 2 columns):
# Column Non-Null Count Dtype
— — — — — — — — — — — — — — -
0 ac 24 non-null int64
1 teach 24 non-null int64
dtypes: int64(2)
memory usage: 512.0 bytes
import statsmodels.api as sm
from statsmodels.formula.api import ols
model = ols(‘ac ~ C(teach)’, data=df).fit()
table = sm.stats.anova_lm(model, typ=2)
print(table)
sum_sq df F PR(>F)
C(teach) 1764.125000 3.0 31.209642 9.677198e-08
Residual 376.833333 20.0 NaN NaN
#The ANOVA has separated components of the sum of squares into two parts, that due to teaching and that leftover in residual. The sum of squares for teaching is equal to 1764.125,
while the residual sums of squares are equal to 376.833. The total sum of squares is therefore equal to 1764.125+376.833=2140.958.
# Each sum of squares is associated with its degrees of freedom. The degree of freedom for teaching is equal to one minus the number of groups. Since there are four groups, the degree of freedom for teaching is equal to 3.0. From within we lose a single degree of freedom per group across all groups. In science, there are four groups which yields a total of
20 degrees of freedom.
#The mean square of ANOVA is not reported in the table, though they are implicit since they are used to compute the resulting F-ratio. The mean squares are computed as SS/df, which in our case for teaching are equal to 376.833/20=18.84.
# The F-statistics for the ANOVArecall is computed as a ratio of MS
between to MS within, so for our data, the computation is 588.04/18.84=31.212, which is evaluated as coming from an F distribution on 3 and 20 degrees of freedom.
#The P-value resulting from the significance test on F is equal to 9.67719.
when translated out scientific notation, equals 0.00000009677198. Hence
we can see it is much less than the conventional level set such as 0.05, and hence we deem the Fstatistic statistically significant., so we have evidence to reject the null hypothesis and conclude that somewhere among the means there is a mean population difference.
#Simply because we have rejected the null, however, does not inform us
of where exactly the mean differences may lie. The F-test is an omnibus
test such that it gives us the overall test but it does not tell us specifically
which means are different from one another. For that, we will require a
post-hoc test.
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