Differences between means (review)

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#from IPython.core.display import HTML, Markdown, display
from IPython.core.display import HTML, Markdown

import numpy.random as npr
import numpy as np
import pandas as pd
import seaborn as sns
import matplotlib.pyplot as plt
import scipy.stats as stats
import statsmodels.formula.api as smf
import pingouin as pg

import ipywidgets as widgets

# Enable plots inside the Jupyter Notebook
%matplotlib inline

2+2

4

Differences between means (review)#

Authored by Todd Gureckis with input from Matt Crump.

Exercise 1: Bootstrapping the \(t\)-distribution#

ts=[]
for _ in range(10000): # repeat 10000 times
    r_sample = np.random.normal(0,1,size=6) #set size size according to instruction
    sem = np.std(r_sample,ddof=1)/np.sqrt(len(r_sample))
    t_stat = np.mean(r_sample)/sem
    ts.append(t_stat)
    
ts2=[]
for _ in range(10000):
    r_sample = np.random.normal(0,1,size=500) #set size according to instructions
    sem = np.std(r_sample,ddof=1)/np.sqrt(len(r_sample))
    t_stat = np.mean(r_sample)/sem
    ts2.append(t_stat)
    
sns.histplot(ts)
sns.histplot(ts2)

Exercise 2: Relationship between \(p\) and \(t\) values#

@widgets.interact(dof=widgets.IntSlider(min=1, max=53, step=1, value=10), alpha=widgets.FloatSlider(min=0,max=0.5, step=0.01, value=0.2))
def plot_t_onsided(dof, alpha):
    fix, ax = plt.subplots(1,1,figsize=(10,6))

    x=np.linspace(-3.5,3.5,100)
    y=stats.t.pdf(x,df=dof)
    t_crit=stats.t.ppf(1.0-alpha, df=dof)
    print(t_crit)
    ax.plot(x,y)
    ax.set_ylabel("probability")
    ax.set_xlabel("value of t statistic")
    ax.set_title("One Sided Test")
    ax.fill_between(x,y,where=x>t_crit,interpolate=True,facecolor='lightblue',alpha=0.2,hatch='/',edgecolor='b')
    ax.set_xticks([0, t_crit])
    #ax.set_yticklabels([])


    sns.despine(top=True, right=True, left=True)

    plt.show()

Exercise 3: Computing a one-sample \(t\)-test by hand#

scores=np.array([.5,.56,.76,.8,.9])  # here is your data
m = scores.mean()
print (m)
sd = scores.std(ddof=1)
print (sd)
N = len (scores)
print (N)
se = sd / np.sqrt (N)
print (se)
t = (m - .25) / se
print (t)
p = 1 - stats.t.cdf (t, N - 1)
print (p)

0.704
0.16816658407662327
5
0.07520638270785267
6.03672166714376
0.0018983152219906874

Exercise 4: Using pingouin to do a one-sample ttest()#

import pingouin as pg
scores=np.array([.5,.56,.76,.8,.9])  # here is your data
pg.ttest(x=scores, y=0.25, alternative='greater')
T dof alternative p-val CI95% cohen-d BF10 power
T-test 6.036722 4 greater 0.001898 [0.54, inf] 2.699704 28.44 0.999169

Exercise 5: Melting data#

exp_df = pd.DataFrame({"subjects": [1,2,3,4,5], "level_A": [1,4,3,6,5], "level_B": [4,8,7,9,10]})
exp_df

tidy_exp_df=exp_df.melt(id_vars='subjects', var_name='level', value_name='test_value')
tidy_exp_df

tidy_exp_df.pivot(index='subjects',columns='level', values='test_value').reset_index()

Exercise 6: Paired-sample \(t\)-test example#

STUDY DESCRIPTION#

Parents often sing to their children and, even as infants, children listen to and look at their parents while they are singing. Research by Mehr, Song, and Spelke (2016) sought to explore the psychological function that music has for parents and infants, by examining the hypothesis that particular melodies convey important social information to infants. Specifically, melodies convey information about social affiliation.

The authors argue that melodies are shared within social groups. Whereas children growing up in one culture may be exposed to certain songs as infants (e.g., “Rock-a-bye Baby”), children growing up in other cultures (or even other groups within a culture) may be exposed to different songs. Thus, when a novel person (someone who the infant has never seen before) sings a familiar song, it may signal to the infant that this new person is a member of their social group.

To test this hypothesis, the researchers recruited 32 infants and their parents to complete an experiment. During their first visit to the lab, the parents were taught a new lullaby (one that neither they nor their infants had heard before). The experimenters asked the parents to sing the new lullaby to their child every day for the next 1-2 weeks.

Following this 1-2 week exposure period, the parents and their infant returned to the lab to complete the experimental portion of the study. Infants were first shown a screen with side-by-side videos of two unfamiliar people, each of whom were silently smiling and looking at the infant.The researchers recorded the looking behavior (or gaze) of the infants during this ‘baseline’ phase. Next, one by one, the two unfamiliar people on the screen sang either the lullaby that the parents learned or a different lullaby (that had the same lyrics and rhythm, but a different melody). Finally, the infants saw the same silent video used at baseline, and the researchers again recorded the looking behavior of the infants during this ‘test’ phase.For more details on the experiment’s methods, please refer to Mehr et al. (2016) Experiment 1.

The first thing to do is download the .csv-formatted data file.

# get the baby data frame
baby_df = pd.read_csv('MehrSongSpelke2016.csv')
# filter to only have the data from experiment 1
experiment_one_df = baby_df[baby_df['exp1']==1]
experiment_one_df.head()
id study_code exp1 exp2 exp3 exp4 exp5 dob dot1 dot2 ... dtword13 dtnoword13 totsing14 babylike14 singcomf14 totrecord14 othersong14 dtword14 dtnoword14 filter_$
0 101 "LUL" 1 0 0 09oct2012 29mar2013 05apr2013 ... 0 0 0 0 1
1 102 "LUL" 1 0 0 16nov2012 10may2013 17may2013 ... 0 0 0 0 1
2 103 "LUL" 1 0 0 26nov2012 11may2013 20may2013 ... 0 0 0 0 1
3 104 "LUL" 1 0 0 19nov2012 11may2013 18may2013 ... 0 0 0 0 1
4 105 "LUL" 1 0 0 29nov2012 15may2013 29may2013 ... 0 0 4 3 4 0 0 0 0 1

5 rows × 153 columns

Baseline phase: Conduct a one-sample \(t\)-test#

You first want to show that infants’ looking behavior did not differ from chance during the baseline trial. The baseline trial was 16 seconds long. During the baseline, infants watched a video of two unfamiliar people, one of the left and one on the right. There was no sound during the baseline. Both of the actors in the video smiled directly at the infant.

The important question was to determine whether the infant looked more or less to either person. If they showed no preference, the infant should look at both people about 50% of the time. How could we determine whether the infant looked at both people about 50% of the time?

The experiment_one_df data frame has a column called Baseline_Proportion_Gaze_to_Singer. All of these values show the proportion of time that the infant looked to the person who would later sing the familiar song to them. If the average of these proportion is .5 across the infants, then we would have some evidence that the infants were not biased at the beginning of the experiment. However, if the infants on average had a bias toward the singer, then the average proportion of the looking time should be different than .5.

Using a one-sample \(t\)-test, we can test the hypothesis that our sample mean for the Baseline_Proportion_Gaze_to_Singer was not different from .5.

# here is how to get the column
x=experiment_one_df['Baseline_Proportion_Gaze_to_Singer']
print (x.mean ())

0.5210966749999999

pg.ttest (x,y=.50)
T dof alternative p-val CI95% cohen-d BF10 power
T-test 0.674375 31 two-sided 0.505071 [0.46, 0.58] 0.119214 0.233 0.100209

Remember how the experiment went. Infants watched silent video recordings of two women (Baseline). Then each person sung a song, one was familiar to the infant (their parents sung the song to them many times), and one was unfamiliar (singing phase). After the singing phase, the infants watched the silent video of the two singers again (test phase). The critical question was whether the infants would look more to the person who sung the familiar song compared to the person who sung the unfamiliar song. If the infants did this, they should look more than 50% of the time to the singer who sang the familiar song. We have the data, we can do another one-sample \(t\)-test to find out.

# here is how to get the column
z=experiment_one_df['Test_Proportion_Gaze_to_Singer']
print (z.mean ())

0.59349125

pg.ttest(z, y=.5)
T dof alternative p-val CI95% cohen-d BF10 power
T-test 2.959714 31 two-sided 0.005856 [0.53, 0.66] 0.523208 6.959 0.817784 
pg.ttest (x, z, paired=True)
T dof alternative p-val CI95% cohen-d BF10 power
T-test -2.41643 31 two-sided 0.021753 [-0.13, -0.01] 0.407103 2.297 0.606906