GG274 Homework 8: Hypothesis Testing

GG274 Homework 8: Hypothesis Testing#

Introduction#

Residential instability is one component of the Ontario Marginalization Index that includes indicators of types and density of residential accommodations, and certain family structure characteristics, such as living alone and dwelling owndership. (see OCHPP)

In this homework you will explore the following question:

Are mental health visits different in Toronto neighbourhoods with higher “residential instability”?

import numpy as np
import pandas as pd

Step 1 - Read the Neighbourhood Instability data into a pandas DataFrame#

a) The data is stored in 1_marg_neighb_toronto_2006_OnMarg.xls - a Microsoft Excel file format with file extension .xls.

Use the pandas function read_excel to read the sheet Neighbourhood_Toronto_OnMarg into a pandas DataFrame named marg_neighbourhoods.

marg_neighbourhoods = pd.read_excel("1_marg_neighb_toronto_2006_OnMarg.xls",
                                    sheet_name="Neighbourhood_Toronto_OnMarg", 
                                    header=1)

marg_neighbourhoods.head()
Neighb id Neighbourhood name POPULATION INSTABILITY INSTABILITY_Q DEPRIVATION DEPRIVATION_Q ETHNICCONCENTRATION ETHNICCONCENTRATION_Q DEPENDENCY DEPENDENCY_Q ONMARG_COMBINED_Q
0 1 West Humber-Clairville 32252 -0.6639 1 0.1620 3 2.4548 5 -0.2021 3 2.4
1 2 Mount Olive-Silverstone-Jamestown 32127 -0.1081 1 1.0195 5 3.7433 5 -0.5975 1 2.4
2 3 Thistletown-Beaumond Heights 9928 -0.3131 1 0.3460 4 1.6220 4 0.2845 5 2.8
3 4 Rexdale-Kipling 10725 0.1866 2 0.4704 4 1.2396 3 0.2734 5 2.8
4 5 Elms-Old Rexdale 9879 -0.0150 2 0.8040 5 1.9911 4 -0.3527 2 2.6

Use marg_neighbourhoods to create a another DataFrame called instability_df that has three columns: "Neighb id ", "Neighbourhood name ", "INSTABILITY".

instability_df = marg_neighbourhoods[marg_neighbourhoods.columns[[0, 1, 3]]]

list(instability_df)

['Neighb id ', 'Neighbourhood name ', 'INSTABILITY']

b) Rename the column names of instability_df using the following table. The DataFrame with the new column names should be called instability_df (i.e., don’t change the name of the DataFrame).

Original column name

New column name

“Neighb id “

n_id

“INSTABILITY”

instability

“Neighbourhood name “

name

Be aware of the trailing spaces!

colnames = {"Neighb id ": "n_id",
            "INSTABILITY" : "instability",
            "Neighbourhood name ": "name"}

instability_df = instability_df.copy()

instability_df.rename(columns=colnames, inplace=True)

instability_df.head()
n_id name instability
0 1 West Humber-Clairville -0.6639
1 2 Mount Olive-Silverstone-Jamestown -0.1081
2 3 Thistletown-Beaumond Heights -0.3131
3 4 Rexdale-Kipling 0.1866
4 5 Elms-Old Rexdale -0.0150

Step 2 - Read the mental health visit data into a pandas DataFrame.#

a) In this step you will read in data on rates of mental health visits stored in 2_ahd_neighb_db_ast_hbp_mhv_copd_2012.xls into a pandas DataFrame named mental_health_neighbourhoods.

mental_health_neighbourhoods = pd.read_excel("2_ahd_neighb_db_ast_hbp_mhv_copd_2012.xls",
                                             sheet_name="2_MentalHealthV_2012", 
                                             header=11)
mental_health_neighbourhoods.head()
Unnamed: 0 Unnamed: 1 Male Female Both sexes Male.1 Female.1 Both sexes.1 Male.2 Female.2 ... Female.12 Both sexes.12 Rate Ratio**, Both sexes.4 H/ L/ NS, Both sexes.4 (95% CI) LL, Male.4 (95% CI) UL, Male.4 (95% CI) LL, Female.4 (95% CI) UL, Female.4 (95% CI) LL, Both sexes.4 (95% CI) UL, Both sexes.4
0 1 West Humber-Clairville 938 1168 2106 13915 14046 27961 6.6 8.2 ... 7.2 7.2 0.85 L 6.1 8.3 6.2 8.3 6.4 8.0
1 2 Mount Olive-Silverstone-Jamestown 866 1130 1996 12256 13082 25338 7.0 8.6 ... 8.5 8.2 0.96 NS 6.6 9.4 7.2 10.0 7.3 9.3
2 3 Thistletown-Beaumond Heights 275 410 685 4124 4453 8577 6.4 9.2 ... 8.1 7.9 0.93 NS 5.9 9.9 6.4 10.0 6.7 9.3
3 4 Rexdale-Kipling 328 453 781 4130 4470 8600 7.7 10.0 ... 9.0 9.1 1.07 NS 7.2 11.6 7.3 11.0 7.8 10.6
4 5 Elms-Old Rexdale 287 396 683 3787 4028 7815 7.4 9.6 ... 8.1 7.5 0.88 NS 4.8 9.2 6.2 10.5 6.1 9.1

5 rows × 81 columns

b) Create a new DataFrame mh_visit_rates by selecting the columns in mental_health_neighbourhoods that corresponds to Neighbourhood ID, Neighbourhood Name, and ‘Age-Standardized rate of Mental Health Visits (2012), All Ages 20+’ rename this column in mh_visit_rates to mh_visit_rates_mf. When you rename this column don’t change the name of the DataFrame mh_visit_rates.

mh_visit_rates = mental_health_neighbourhoods[mental_health_neighbourhoods.columns[[0, 1, 10]]]

colnames = {"Unnamed: 0": "n_id",
            "Both sexes.2" : "mh_visit_rates_mf",
            "Unnamed: 1" : "name"}

mh_visit_rates = mh_visit_rates.copy()

mh_visit_rates.rename(columns=colnames, inplace=True)

mh_visit_rates.head()
n_id name mh_visit_rates_mf
0 1 West Humber-Clairville 7.4
1 2 Mount Olive-Silverstone-Jamestown 7.8
2 3 Thistletown-Beaumond Heights 7.8
3 4 Rexdale-Kipling 8.9
4 5 Elms-Old Rexdale 8.5

Step 3 - Merge mental health visits and instability#

In this step you will merge the mh_visit_rates with mental_health_neighbourhoods.

a) Merge mh_visit_rates with instability_df by matching the values of "n_id" and "name", and name this DataFrame mh_visit_instability.

mh_visit_instability = mh_visit_rates.merge(instability_df, on=["n_id", "name"])
mh_visit_instability.head()
n_id name mh_visit_rates_mf instability
0 1 West Humber-Clairville 7.4 -0.6639
1 2 Mount Olive-Silverstone-Jamestown 7.8 -0.1081
2 3 Thistletown-Beaumond Heights 7.8 -0.3131
3 4 Rexdale-Kipling 8.9 0.1866
4 5 Elms-Old Rexdale 8.5 -0.0150

Step 4#

a) Create a new column in mh_visit_instability named instability_binary that categorizes neighbourhoods. The new columns should have two possible values:

  • "High", if the neighbourhood’s instability value is greater than or equal to the mean of "instability"

  • "Low", if the neighbourhood’s instability value is less than the mean of "instability"

mean_instab = mh_visit_instability["instability"].mean()

mh_visit_instability.loc[
    mh_visit_instability["instability"] >= mean_instab, 
    "instability_binary"] = "High"

mh_visit_instability.loc[
    mh_visit_instability["instability"] < mean_instab, 
    "instability_binary"] = "Low"

b) Compute the frequency distribution of instability_binary. Save the results in instability_binary_frequencies.

instability_binary_frequencies = mh_visit_instability["instability_binary"].value_counts()

instability_binary_frequencies

instability_binary
Low     70
High    66
Name: count, dtype: int64

c) Is there evidence that Toronto has many neighbourhoods that have residential instability? Briefly explain. (1 mark)

There is more than one possible way to answer the question. e.g.,

i. The data shows that about half of the neighbourhoods in Toronto, 66 out of 136, having residential instability measures that are higher than the city-wide mean. It is difficult to provide a definite answer to the question based on the result since it only provide comparison within the city.

ii. (a bit more naive answer) About half of the neighbourhoods in Toronto have high residential instability.

Any answer with a sensible reasoning based on the data is acceptable.

Step 5 - Do neighbourhoods with high residential instability have more mental health visits compared to neighbourhoods with low residential isntability?#

a) Use the DataFrame describe method to compute the distribution of mh_visit_rates_mf in mh_visit_instability grouped by instability_binary. Store the results in median_table.

median_table = mh_visit_instability.groupby("instability_binary")["mh_visit_rates_mf"].describe()

median_table
count mean std min 25% 50% 75% max
instability_binary
High 66.0 8.362121 0.937388 5.9 7.825 8.45 9.075 10.5
Low 70.0 7.867143 1.000539 5.7 7.025 7.80 8.775 9.9

Use median_table to compute the difference in medians between neighbourhoods with high and low instability. Store this value in median_diff.

median_diff = median_table["50%"][0] - median_table["50%"][1]

median_diff

C:\Users\yukiz\AppData\Local\Temp\ipykernel_26372\2073326976.py:1: FutureWarning: Series.__getitem__ treating keys as positions is deprecated. In a future version, integer keys will always be treated as labels (consistent with DataFrame behavior). To access a value by position, use `ser.iloc[pos]`
  median_diff = median_table["50%"][0] - median_table["50%"][1]

0.6499999999999995

Step 6 - Set up a simulation in python to test if the medians are equal#

a) In this step you will write a function random_shuffle_median that returns a simulated value of the median difference (a simulated value of the test statistic) of mental health visit rates in neighbourhoods with high versus low residential instability assuming that there really is no difference in mental health visit rates between these types of neighbourhoods.

A step-by-step explantion of a similar function was given in lecture, and you can follow this example to help guide you through this step.

The function random_shuffle_median is started for you below. Your task is to complete the function by filling in the ....

Try writing a meaningful docstring for random_shuffle_median. The pandas docstring guide has some great examples and guidelines. (NB: this will not be graded)

# def random_shuffle_median():
#     """
#     Put your docstring here (optional)
#     """

#     # shuffle the column of mh_visit_instability that corresponds to high/low instability

#     instability_binary_shuffle = mh_visit_instability[...].sample(
#         frac=1, replace=False).reset_index(drop=True)
    
#     # calculate the median visit rate for high and low instability neighbourhoods

#     visitrate_low_shuffle = mh_visit_instability.loc[instability_binary_shuffle == ..., ...].median()
    
#     visitrate_high_shuffle  = mh_visit_instability.loc[instability_binary_shuffle ==  ..., ...].median()
    
#     shuffled_diff = ... - ...
    
#     return shuffled_diff

def random_shuffle_median():
    """
    Randomly shuffles the column of mh_visit_instability that corresponds to 

    high/low instability and calculates the difference in median visit rates 
    
    for high and low instability neighbourhoods
    """

    instability_binary_shuffle = mh_visit_instability["instability_binary"].sample(
        frac=1, replace=False).reset_index(drop = True)
    visitrate_low_shuffle = mh_visit_instability.loc[
        instability_binary_shuffle == "Low", "mh_visit_rates_mf"].median()
    visitrate_high_shuffle  = mh_visit_instability.loc[
        instability_binary_shuffle == "High", "mh_visit_rates_mf"].median()
    shuffled_diff = visitrate_high_shuffle - visitrate_low_shuffle 
    return shuffled_diff

random_shuffle_median()

0.0

print(random_shuffle_median.__doc__)

    Randomly shuffles the column of mh_visit_instability that corresponds to 

    high/low instability and calculates the difference in median visit rates 
    
    for high and low instability neighbourhoods

b) Explain the purpose of

mh_visit_instability[...].sample(
    frac=1, replace=False).reset_index(drop=True)

in 1-2 sentences.

The line randomly samples 100% (frac=1) of the selected column ('instability_binary') without replacements (replace=False) and removes the old ordering (reset_index(drop=True)). It mimicks random asingment of the high vs low labels to each neighbourhood.

Step 7 - Compute the distribution of simulated values of the median difference assuming the null hypothesis is true#

We will use your student number to generate data for this homework. Complete the assignment statement below by typing your student number as an int. In other words assign your student number as an integer to the variable student_number.

# # Replace the ... with your student number
student_number = 12345

# # This checks that you correctly typed in your student_number as an int.
# # Make sure there's no error when you run this cell!
# assert type(student_number) == int

a) Write a function called shuffled_diffs that returns a list. The function should use a for loop that iterates the function random_shuffle_median an arbitrary number of times. The number of times that the for loop iterates should be controlled by a function parameter named number_of_shuffles.

def shuffled_diffs(number_of_shuffles):
    shuffled_diffs = []
    for _ in range(number_of_shuffles):
        shuffled_diffs.append(random_shuffle_median())
    return shuffled_diffs

b) Use shuffled_diffs to compute 10 000 simulated median differences between high and low instability neighbourhoods assuming that there is no difference in median mental health visit rates between high and low instability neighbourhoods. Store the values in shuffled_diffs_10000.

np.random.seed(student_number)

shuffled_diffs_10000 = shuffled_diffs(10000)

c) Plot the distribution of the 10,000 simulated values stored in shuffled_diffs_10000 using a matplotlib histogram. Name the plot null_hypothesis_distribution_plot. Label the horizontal axis as "Difference in median visit rates for high and low instability neighbourhoods" and the vertical axis as "Frequency".

import matplotlib.pyplot as plt

null_hypothesis_distribution_plot = plt.hist(shuffled_diffs_10000)

plt.xlabel("Difference in median visit rates for high and low instability neighbourhoods")

plt.ylabel("Frequency")

Text(0, 0.5, 'Frequency')
../../../../_images/5a71108a48d5f34e147406dd5eae0c1bd94e0a7137a773a5ea46d521534b0082.png

Step 8 - Compute the p-value#

a) Compute the number of simulated differences in medians in shuffled_diffs_10000 that are greater than or equal to the observed median difference (median_diff). Store this value in right_extreme.

right_extreme = shuffled_diffs_10000 >= median_diff

right_extreme = right_extreme.sum()

b) Compute the number of simulated differences in medians in shuffled_diffs_10000 that are less than the observed median difference (median_diff). Store this value in left_extreme.

left_extreme = shuffled_diffs_10000 < -median_diff

left_extreme = left_extreme.sum()

c) Use right_extreme and left_extreme to compute the p-value. Store the p-value in pvalue.

pvalue = (left_extreme + right_extreme) / 10000

pvalue

0.0064

Step 9 - Communicate what you did in the steps above#

a) In a few sentences introduce the question that you explored in this homework (see the beginning of this homework). For example, why do you or others think this is an important question? (1 mark)

We investigated whether neighbourhoods whose residential statuses are relatively instable visit mental health services more or less frequently compared to those with relatively stable residential statuses. The analysis may tell us whether the marginalized neighbourhoods need more support for their mental health.

b) Briefly describe the data sources that you used to answer the question. Which statistical variables did you use and why did you use these varaibles? (1 mark)

We used the residential instabilty measure from the Ontario Marginalization Index and the age-standardized rates for mental health visits among those above age 20. We computed the mean of the instability measure to dicotomize the neighbourhoods to those whose measures are above the mean and those whose measures are below the mean. This allowed us to label the neighbourhoods into two groups - high vs. low residential instability.

We then computed the difference in the median mental health visit rates between the two groups. The medians provided the “typical” (or centre of the distributions) visit rates of the two groups.

c) What computational and statistical methods or analyses did you use to answer the question? Briefly describe these methods and how they were used to answer the question. (1 mark)

We simulated the distribution of the difference in median mental health visits rates under the assumption that there is no difference in the median rates betwee the two groups using the bootstrap. We then located the observed difference in the simulated distribution to assess high likely it would be to observe the value when there is actually no difference. When the location of the value in the distribution indicates that it is unlikely, we can reasonably conclude that the median mental health visit rates are different between the two groups.

Note: Full marks if the data and method used are reasonably well explained between part b) and c).

d) Briefly describe the results of your statistical analysis in a few sentences. (1 mark)

After locating the observed difference in the simulated distribution, the resulting p-value is 0.0064 which is very small. It is reasonable to reject the hypothtesis that there is no difference between the two groups and conclude that the mental health visit rates are different between high and low residential instability neighbourhoods. Specifically, the visit rates are higher for those in neighbourhood with higher residential instability.

e) What conclusions can you draw about the question you set out to answer that is supported by the data and statistical analysis of the data? State at least one limitation of your conclusions. (See the USC Research Guide section on study limitations) (1 mark)

The data shows that the typical mental health visit rates in Toronto neighbourhoods with relatively unstable residential statuses is higher. Our statistical analysis provides a reasonably strong evidence that there is a systematic difference and we didn’t observe the difference by chance.

Limitations may include


Any reasonable limitation identified is acceptable.

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