Do Left-handed People Really Die Young?

A 1991 study reported that left-handed people die on average nine years earlier than right-handed people.
Let's explore this phenomenon using age distribution data to see if we can reproduce a difference in average age at death purely from the changing rates of left-handedness over time, refuting the claim of early death for left-handers.

Datasets

• Death distribution data for the United States from the year 1999
• Rates of left-handedness

1. Where are the old left-handed people?

A National Geographic survey in 1986 resulted in over a million responses that included age, sex, and hand preference for throwing and writing. Researchers Avery Gilbert and Charles Wysocki analyzed this data and noticed that rates of left-handedness were around 13% for people younger than 40 but decreased with age to about 5% by the age of 80. They concluded based on analysis of a subgroup of people who throw left-handed but write right-handed that this age-dependence was primarily due to changing social acceptability of left-handedness.
Let's start by plotting the rates of left-handedness as a function of age.

data_url_1 = "https://gist.githubusercontent.com/mbonsma/8da0990b71ba9a09f7de395574e54df1/raw/aec88b30af87fad8d45da7e774223f91dad09e88/lh_data.csv"
lefthanded_data = pd.read_csv(data_url_1)

fig, ax = plt.subplots() # create figure and axis objects
ax.plot(lefthanded_data.Age, lefthanded_data.Female,label='Female', marker = 'o') # plot "Female" vs. "Age"
ax.plot(lefthanded_data.Age, lefthanded_data.Male, label='Male', marker = 'x') # plot "Male" vs. "Age"

plt.show()

2. Rates of left-handedness over time

Let's convert this data into a plot of the rates of left-handedness as a function of the year of birth, and average over male and female to get a single rate for both sexes.
Since the study was done in 1986, the data after this conversion will be the percentage of people alive in 1986 who are left-handed as a function of the year they were born.

lefthanded_data['Birth_year'] = 1986 - lefthanded_data.Age
lefthanded_data['Mean_lh'] = (lefthanded_data.Male+lefthanded_data.Female)/2

fig, ax = plt.subplots()
ax.plot(lefthanded_data.Birth_year, lefthanded_data.Mean_lh) # plot 'Mean_lh' vs. 'Birth_year'

plt.show()

3. Applying Bayes' rule

We want to calculate the probability of dying at age A given that you're left-handed.
Here's Bayes' theorem for the two events we care about: left-handedness LH and dying at age A.

• P(LH | A) the probability that you are left-handed given that you died at age A
• P(A) overall probability of dying at age A
• P(LH) overall probability of being left-handed

To calculate P(LH | A) for ages that might fall outside the original data, we will need to extrapolate the data to earlier and later years.

def P_lh_given_A(ages_of_death, study_year = 1990):
    """ P(Left-handed | age of death), calculated based on the reported rates of left-handedness.
    Inputs: age of death, study_year
    Returns: probability of left-handedness given that a subject died in `study_year` at age `age_of_death` """
    
    # Use the mean of the 10 neighbouring points for rates before and after the start 
    early_1900s_rate = lefthanded_data['Mean_lh'][-10:].mean()
    late_1900s_rate = lefthanded_data['Mean_lh'][:10].mean()
    middle_rates = lefthanded_data.loc[lefthanded_data['Birth_year'].isin(study_year - ages_of_death)]['Mean_lh']
    
    youngest_age = study_year - 1986 + 10 # the youngest age in the NatGeo dataset is 10
    oldest_age = study_year - 1986 + 86 # the oldest age in the NatGeo dataset is 86
    
    P_return = np.zeros(ages_of_death.shape)  # create an empty array to store the results
    # extract rate of left-handedness for people of age age_of_death
    P_return[ages_of_death > oldest_age] = early_1900s_rate / 100
    P_return[ages_of_death < youngest_age] = late_1900s_rate / 100
    P_return[np.logical_and((ages_of_death <= oldest_age), (ages_of_death >= youngest_age))] = middle_rates / 100
 
    return P_return

4. When do people normally die? P(A)

To estimate the probability of living to an age A, we can use data that gives the number of people who died in a given year and how old they were to create a distribution of ages of death.

# Death distribution data for the United States in 1999
data_url_2 = "https://gist.githubusercontent.com/mbonsma/2f4076aab6820ca1807f4e29f75f18ec/raw/62f3ec07514c7e31f5979beeca86f19991540796/cdc_vs00199_table310.tsv"
death_distribution_data = pd.read_csv(data_url_2, sep='\t', skiprows=[1])

death_distribution_data = death_distribution_data.dropna(subset=['Both Sexes'])

fig, ax = plt.subplots()
ax.plot(death_distribution_data['Age'], death_distribution_data['Both Sexes']) # plot 'Both Sexes' vs. 'Age'

plt.show()

5. The overall probability of left-handedness P(LH)

P(LH) is the probability that a person who died in our particular study year is left-handed. We can calculate it by summing up all of the left-handedness probabilities for each age, weighted with the number of deceased people at each age, then divided by the total number of deceased people to get a probability.

• N(A) is the number of people who died at age A

def P_lh(death_distribution_data, study_year = 1990): # sum over P_lh for each age group
    """ Overall probability of being left-handed if you died in the study year
    P_lh = P(LH | Age of death) P(Age of death) + P(LH | not A) P(not A) = sum over ages 
    Input: dataframe of death distribution data
    Output: P(LH), a single floating point number """
    p_list = death_distribution_data['Both Sexes']*P_lh_given_A(death_distribution_data['Age'], study_year)
    p = np.sum(p_list)
    return p/np.sum(death_distribution_data['Both Sexes']) # normalize to total number of people in distribution

>>> P_lh(death_distribution_data)
... 0.07766387615350638

6. Putting it all together: dying while left-handed (i)

Now we have the means of calculating all three quantities we need: P(A), P(LH) and P(LH | A)
We can combine all three using Bayes' rule to get P(A | LH)
.

For left-handers,

def P_A_given_lh(ages_of_death, death_distribution_data, study_year = 1990):
    """ The overall probability of being a particular `age_of_death` given that you're left-handed """
    P_A = death_distribution_data['Both Sexes'][ages_of_death] / np.sum(death_distribution_data['Both Sexes'])
    P_left = P_lh(death_distribution_data, study_year) # use P_lh function to get probability of left-handedness overall
    P_lh_A = P_lh_given_A(ages_of_death, study_year) # use P_lh_given_A to get probability of left-handedness for a certain age
    return P_lh_A*P_A/P_left

And for right-handers,

def P_A_given_rh(ages_of_death, death_distribution_data, study_year = 1990):
    """ The overall probability of being a particular `age_of_death` given that you're right-handed """
    P_A = death_distribution_data['Both Sexes'][ages_of_death] / np.sum(death_distribution_data['Both Sexes'])
    P_right = 1- P_lh(death_distribution_data, study_year) # either you're left-handed or right-handed, so these sum to 1
    P_rh_A = 1-P_lh_given_A(ages_of_death, study_year) # these also sum to 1 
    return P_rh_A*P_A/P_right

8. Plotting the distributions of conditional probabilities

Let's plot these probabilities for a range of ages of death from 6 to 120.

ages = np.arange(6,115,1)

left_handed_probability = P_A_given_lh(ages, death_distribution_data)
right_handed_probability = P_A_given_rh(ages, death_distribution_data)

fig, ax = plt.subplots() # create figure and axis objects
ax.plot(ages, left_handed_probability, label = "Left-handed")
ax.plot(ages, right_handed_probability, label = "Right-handed")

plt.show()

Notice that the left-handed distribution has a bump below age 70: of the pool of deceased people, left-handed people are more likely to be younger.

9. Moment of truth: age of left and right-handers at death

Finally, let's compare our results with the original study that found that left-handed people were nine years younger at death on average. We can do this by calculating the mean of these probability distributions in the same way we calculated P(LH) earlier, weighting the probability distribution by age and summing over the result.

average_lh_age =  np.nansum(ages*np.array(left_handed_probability))
average_rh_age =  np.nansum(ages*np.array(right_handed_probability))


>>> print(round(average_lh_age,1))
... 67.2
>>> print(round(average_rh_age,1))
... 72.8
>>> print("The difference in average ages is " + str(round(average_rh_age - average_lh_age, 1)) + " years.")
... The difference in average ages is 5.5 years.

10. Final comments

We got a pretty big age gap between left-handed and right-handed people purely as a result of the changing rates of left-handedness in the population.
To finish off, let's calculate the age gap we'd expect if we did the study in 2021 instead of in 1990.

left_handed_probability_2021 = P_A_given_lh(ages, death_distribution_data, study_year = 2021)
right_handed_probability_2021 = P_A_given_rh(ages, death_distribution_data, study_year = 2021)
    
average_lh_age_2021 =  np.nansum(ages*np.array(left_handed_probability_2021))
average_rh_age_2021 =  np.nansum(ages*np.array(right_handed_probability_2021))

>>> print("The difference in average ages is " + str(round(average_rh_age_2021 - average_lh_age_2021, 1)) + " years.")
... The difference in average ages is 1.9 years.