3.7. Independent samples t-test#

3.7.1. Set up Python libraries#

As usual, run the code cell below to import the relevant Python libraries

# Set-up Python libraries - you need to run this but you don't need to change it
import numpy as np
import matplotlib.pyplot as plt
import scipy.stats as stats
import pandas as pd
import seaborn as sns
sns.set_theme(style='white')
import statsmodels.api as sm
import statsmodels.formula.api as smf
import warnings 
warnings.simplefilter('ignore', category=FutureWarning)

3.7.2. Example#

https://raw.githubusercontent.com/jillxoreilly/StatsCourseBook/main/images/appletree.jpg

Below we have data on the average weight, in grams, of apples from each of 20 apple trees.

10 of the trees received MiracleGro fertilizer and the other 10 received Brand X.

Test the hypothesis that the trees given MiracleGro produced heavier apples.

Inspect the data#

The data are provided in a text (.csv) file.

Let’s load the data as a Pandas dataframe, and plot them to get a sense for their distribution (is it normal?) and any outliers

# load the data and have a look
apples = pd.read_csv('https://raw.githubusercontent.com/jillxoreilly/StatsCourseBook_2024/main/data/AppleWeights.csv')
apples
Fertilizer meanAppleWeight
0 BrandX 172
1 BrandX 165
2 BrandX 175
3 BrandX 164
4 BrandX 165
5 BrandX 157
6 BrandX 183
7 BrandX 186
8 BrandX 191
9 BrandX 173
10 MiracleGro 164
11 MiracleGro 198
12 MiracleGro 184
13 MiracleGro 200
14 MiracleGro 180
15 MiracleGro 189
16 MiracleGro 177
17 MiracleGro 170
18 MiracleGro 192
19 MiracleGro 193
20 MiracleGro 187
21 MiracleGro 176

Let’s plot the data and see if they look Normally distributed.

As we saw in the session on plotting, a good choice here will be a KDE plot (to get an estimate of the shape of the distribution) and a rug plot (individual data values as the KDE plot is based on only a small sample)

sns.kdeplot(data=apples, x='meanAppleWeight', hue='Fertilizer', fill=True)
sns.rugplot(data=apples, x='meanAppleWeight',  hue='Fertilizer', height=0.1)
plt.xlabel("Mean weight of apples (g)", fontsize = 12)
plt.ylabel("Density", fontsize = 12)

plt.show()
../_images/e65b55eba1298df5d0e851a9de05f59e431b764be4ddc78ca40c39aef45b6c49.png

It looks like both distributions are unimodal but a bit skewed.

Typically, it would be hard to say if the data are really non-normal due to the small number of data points (the apparent skew could be a fluke). By default, I would err on the side of caution and use a non-parametric test.

However, in this particular case I am confident that the data in each sample are drawn from a normal distribution because each data point represents the mean of a large sample (the weights of all apples on one tree) and such means are always normally distributed due to the Central Limit theorem. There is a video explaining why here.

So we can go ahead and use a t-test.

Hypotheses#

\(\mathcal{H_o}\): the mean weight of apples is the same for trees fertilized with MiracleGro and Brand X

\(\mathcal{H_a}\): the mean weight of apples is greater for trees fertilized with MiracleGro

This is a one tailed test as the manufacturers of MiracleGro are only looking for an effect in one direction (evidence that MiracleGro is better!)

We will test at the \(\alpha = 0.05\) significance level

Descriptive statistics#

First, we obtain the relevant desriptive statistics. By relevant, I mean the ones that go into the equation for the t-test:

\[ t = \frac{\bar{x_1} - \bar{x_2}}{s \sqrt{\frac{1}{n_1}+\frac{1}{n_2}}} \]

This would be the sample means \(\bar{x_1}\) and \(\bar{x_2}\), and the standard deviations for both samples (these feed into the pooled standard deviation \(s\)) and the sample size \(n\).

Remember, apples is our original Pandas dataframe:

display(apples)
Fertilizer meanAppleWeight
0 BrandX 172
1 BrandX 165
2 BrandX 175
3 BrandX 164
4 BrandX 165
5 BrandX 157
6 BrandX 183
7 BrandX 186
8 BrandX 191
9 BrandX 173
10 MiracleGro 164
11 MiracleGro 198
12 MiracleGro 184
13 MiracleGro 200
14 MiracleGro 180
15 MiracleGro 189
16 MiracleGro 177
17 MiracleGro 170
18 MiracleGro 192
19 MiracleGro 193
20 MiracleGro 187
21 MiracleGro 176

We obtain some commonly used descriptive statistics using the df.describe() method in pandas

apples.describe()
meanAppleWeight
count 22.000000
mean 179.136364
std 12.123552
min 157.000000
25% 170.500000
50% 178.500000
75% 188.500000
max 200.000000

We need the descriptive statistics separately for each fertilizer type.

We could use the separate dataframes that we created for plotting, but pandas has a handy method called df.groupby() that will do the job:

apples.groupby(["Fertilizer"]).describe()
meanAppleWeight
count mean std min 25% 50% 75% max
Fertilizer
BrandX 10.0 173.100000 10.867382 157.0 165.00 172.5 181.00 191.0
MiracleGro 12.0 184.166667 11.101460 164.0 176.75 185.5 192.25 200.0

It does look like the mean weight of apples from the MiracleGro trees is higher, but is the difference statistically significant?

Carry out the test#

We carry out an independent samples t-test using the function ttest_ind from scipy.stats, here loaded as stats

stats.ttest_ind(apples.query('Fertilizer=="MiracleGro"').meanAppleWeight, 
                apples.query('Fertilizer=="BrandX"').meanAppleWeight,
                alternative='greater')
Ttest_indResult(statistic=2.350347501385599, pvalue=0.014564862730138283)

The inputs to stats.ttest are the two samples to be compared (the values in the meanAppleWeight column from our separated Pandas data frames apples_BrandX and apples_MiracleGro) and the argument alternative=’greater’, which tells the computer to run a one tailed test that mean of the first input apples.MiracleGro is greater than the second apples.BrandX.

The outputs are statistic (\(t=2.35\)) and pvalue (\(p=0.0146\)) - if this is less than our \(\alpha\) value 0.5, there is a significant difference.

Degrees of freedom#

In a scientific write-up we also need to report the degrees of freedom of the test. This tells us how many observations (data-points) the test was based on, corrected for the number of means we had to estimate from the data in order to do the test.

In the case of the independent samples t-test \(df = n_1 + n_2 - 2\) so in this case, df=(10+12-2)=20 and we can report out test results as:

\(t(20) = 2.35, p=0.0146\) (one-tailed)

Interpretation#

Our t value of 2.35 means that the difference in mean apple weights between MiracleGro and BrandX trees is 2.35 times the standard error (where \( SE = s \sqrt{\frac{1}{n_1}+\frac{1}{n_2}}\)).

Such a large difference (in the expected direction) would occur 0.0146 (1.46%) of the time due to chance if the null hypothesis were true (if MiracleGro was really no better than Brand X), hence the p value of 0.0146.

This diagram shows the expected distribution of t-values if the null were true, with our obtained t-value marked:

There should be a picture of the t-distribution here

Draw conclusions#

As p<0.05 we conclude that the mean weight of apples on trees fertilized with MiracleGro is indeed greater than the mean weight of apples on trees fertilized with Brand X

3.7.3. Write-up#


Above, I walked you through how to run the t-test and why we make different choices.

In this section we revisit the analysis, but here we practice writing up our analysis in the correct style for a scientific report.

Replace the XXXs with the correct values!


We tested the hypothesis that the mean weight of apples produced by trees fertilized with MiracleGro was higher than for trees fertilized with Brand X.

The mean apple weight was measured for each of XX trees fertilized with MiracleGro (mean of mean apple weights over XX trees XXX.Xg, sd of mean apple weights over 100 trees, XXX.Xg) and XX trees fertilized with Brand X (mean over XX trees XXX.Xg, sd XXX.Xg).

apples = pd.read_csv('https://raw.githubusercontent.com/jillxoreilly/StatsCourseBook_2024/main/data/AppleWeights.csv')
apples.groupby(["Fertilizer"]).describe()
meanAppleWeight
count mean std min 25% 50% 75% max
Fertilizer
BrandX 10.0 173.100000 10.867382 157.0 165.00 172.5 181.00 191.0
MiracleGro 12.0 184.166667 11.101460 164.0 176.75 185.5 192.25 200.0

Theoretical considerations suggest that data for each group of trees should be drawn from a normal distribution: as individual data points were themselves the means of large samples (all apples from a given tree), these data points should follow a normal distribubtion due to the Central Limit Theorem. This was supported by a plot of the data:

sns.kdeplot(data=apples, x='meanAppleWeight', hue='Fertilizer', fill=True)
sns.rugplot(data=apples, x='meanAppleWeight',  hue='Fertilizer', height=0.1)
plt.xlabel("Mean weight of apples (g)", fontsize = 12)
plt.ylabel("Density", fontsize = 12)

plt.show()
../_images/e65b55eba1298df5d0e851a9de05f59e431b764be4ddc78ca40c39aef45b6c49.png

An independent samples t-test was therefore used to compare the means (alpha = XXX, XXX-tailed).

stats.ttest_ind(apples.query('Fertilizer=="MiracleGro"').meanAppleWeight, 
                apples.query('Fertilizer=="BrandX"').meanAppleWeight,
                alternative='greater')
Ttest_indResult(statistic=2.350347501385599, pvalue=0.014564862730138283)

The weight of apples from the trees fertilized with MiracleGro was indeed found to be significantly higher: t(18) = 2.35, p=0.0146.

3.7.4. Exercises#

  1. Can you work out how to run a two-tailed test on the data?

  1. What happens to the p value if you run a two tailed test instead of one tailed? Is it more or less significant? Why?