T-test, ANOVA and Chi Squared test made easy.

Summary

This educational video explores various statistical tests and their appropriate applications. It delves into the t-test, covering single sample, two-tailed, one-tailed, and paired tests, using real-world examples to elucidate the concepts. The video then moves on to ANOVA, demonstrating how it allows for comparing means across three or more populations. Additionally, the chi-squared test is introduced, addressing both goodness of fit and test of independence scenarios, enabling analyses of categorical variables and proportions. Throughout the video, the presenter emphasizes the importance of understanding the research question and selecting the appropriate statistical test accordingly, making the material highly accessible and engaging.

Takeaways

• 🔑 The key to understanding statistical tests is to first understand the question being asked.
• 👤 For t-tests, we are examining the difference in means between two populations, or between one population at different points in time.
• 🤔 The null hypothesis assumes there is no difference in means, and we calculate the probability of observing our sample data if the null hypothesis were true.
• ✅ If this probability is below a predetermined threshold (typically 0.05), we reject the null hypothesis and conclude the observed difference is statistically significant.
• 📐 There are different types of t-tests: single sample, two-tailed, one-tailed, and paired t-tests, depending on the specific question being asked.
• ⚖️ ANOVA (Analysis of Variance) is used when comparing means across three or more populations.
• 🔍 After an ANOVA identifies a significant difference, multiple comparisons can pinpoint which populations differ from each other.
• 📊 Chi-squared tests are used to test for differences in proportions of categorical variables.
• ✔️ The chi-squared goodness-of-fit test examines if observed proportions match expected proportions.
• 🔗 The chi-squared test of independence checks if the proportions of one categorical variable depend on the values of another variable.

Q & A

What are the three main statistical tests covered in the video?

The three main statistical tests covered in the video are the t-test, ANOVA (analysis of variance), and the chi-squared test.

What is the purpose of the t-test?

The t-test is used to test the difference in means or averages between two populations, between one population at different points in time, or between a sample mean and a presumed or hypothesized mean.

What is the difference between a one-tailed and two-tailed t-test?

A two-tailed t-test is used when we are asking if there is a difference in any direction, while a one-tailed t-test is used when we are asking if there is a difference in a particular direction.

When is a paired t-test used?

A paired t-test is used when there are paired observations in each population, meaning that for each observation in one population, there is a corresponding observation in the other population.

What is the purpose of ANOVA?

ANOVA (analysis of variance) is used to compare the means of three or more populations. It tests the null hypothesis that there are no differences in the means of these populations.

How can you determine which specific populations are driving the difference in means after conducting an ANOVA?

After conducting an ANOVA and rejecting the null hypothesis, a multiple comparison of means (such as Tukey's test) can be performed to identify which specific populations have statistically significant differences in their means.

What is the chi-squared goodness of fit test used for?

The chi-squared goodness of fit test is used to determine if the observed proportions of a categorical variable are significantly different from the expected proportions based on a hypothesized distribution.

What is the purpose of the chi-squared test of independence?

The chi-squared test of independence is used to determine if there is a significant relationship between two categorical variables, i.e., if the proportions of one variable are independent of the other variable.

What is the null hypothesis being tested in a chi-squared test?

In a chi-squared test, the null hypothesis is that there is no difference in proportions (for goodness of fit) or that the variables are independent of each other (for test of independence).

How is the decision to reject or accept the null hypothesis made in hypothesis testing?

The decision to reject or accept the null hypothesis is made by comparing the calculated p-value to a predetermined significance level (typically 0.05). If the p-value is less than the significance level, the null hypothesis is rejected, indicating a statistically significant result.

Outlines

00:00

🎥 Introduction to Statistical Tests

This paragraph introduces the topic of the video, which is an explanation of different statistical tests: t-test, ANOVA (Analysis of Variance), and chi-squared test. It sets the context by stating that understanding the question being asked is crucial to determining the appropriate test and interpreting the results correctly. The paragraph also provides an overview of the specific types of t-tests (single sample, two-tailed, one-tailed, and paired) and chi-squared tests (goodness of fit and test of independence) that will be covered.

05:02

✏️ Exploring the T-Test

This paragraph delves into the details of the t-test, using real data and examples to illustrate different scenarios and applications. It explains that the t-test is used to analyze the difference in means or averages between two populations, one population at different points in time, or a sample mean compared to a hypothesized or presumed average. The concept of hypothesis testing is introduced, where the null hypothesis assumes no difference in means, and the alternative hypothesis states that the observed difference is statistically significant. The paragraph walks through the process of determining statistical significance, considering the p-value and predetermined threshold (often 0.05).

10:02

📊 ANOVA and Chi-Squared Tests

This paragraph covers two additional statistical tests: ANOVA (Analysis of Variance) and the chi-squared test. For ANOVA, it explains that it is used when comparing the means of three or more populations, with the null hypothesis being no difference in means across populations. The paragraph uses real data and visualizations to illustrate the ANOVA process and interpretation of results, including multiple comparisons and confidence intervals. For the chi-squared test, the paragraph introduces the goodness of fit test and the test of independence, both used to analyze categorical variables and proportions. It explains the process of hypothesis testing, setting the null hypothesis of no difference in proportions, and using the p-value and predetermined threshold to determine statistical significance.

Keywords

💡T-test

The t-test is a statistical hypothesis test used to determine if there is a significant difference between the means of two groups or samples. In the video, the t-test is explained as a method to assess if the difference between means from a sample and a presumed population mean, or between two population means, is statistically significant. Examples given include comparing life expectancy in Africa to a presumed mean, and comparing life expectancies between Africa and Europe.

💡Null hypothesis

The null hypothesis is the default assumption or statement being tested in a statistical analysis. It typically represents a claim of no difference or no effect. In the context of the video, the null hypothesis is that there is no difference between the means or proportions being compared. For example, the null hypothesis could be that the life expectancy in Africa and Europe is the same, or that the proportions of small, medium, and large flowers are equal.

💡P-value

A p-value is a probability value used in statistical hypothesis testing to determine the significance of the results. It represents the probability of obtaining results as extreme as the observed data, assuming the null hypothesis is true. In the video, the p-value is compared to a predetermined threshold (often 0.05) to decide whether to reject or fail to reject the null hypothesis. A small p-value (less than the threshold) suggests that the observed difference is statistically significant.

💡ANOVA (Analysis of Variance)

ANOVA is a statistical method used to analyze the differences among the means of three or more independent groups or populations. The video explains that when comparing more than two means, ANOVA is used instead of the t-test. The null hypothesis in ANOVA is that all group means are equal, and the alternative hypothesis is that at least one group mean differs from the others. ANOVA is demonstrated using data on life expectancy across Europe, the Americas, and Asia.

💡Chi-squared test

The chi-squared test is a statistical test used to evaluate the goodness of fit between observed data and expected data, or to assess the independence of two categorical variables. The video covers two types of chi-squared tests: the goodness of fit test (comparing observed proportions to expected proportions) and the test of independence (determining if two categorical variables are related or independent). These tests are demonstrated using examples of flower species and sizes.

💡Hypothesis testing

Hypothesis testing is a systematic method for making statistical decisions using data. It involves formulating a null hypothesis (the default assumption) and an alternative hypothesis, collecting data, and using statistical tests to determine whether the null hypothesis should be rejected or not rejected based on the observed data and a predetermined significance level. The video extensively discusses hypothesis testing in the context of various statistical tests (t-test, ANOVA, chi-squared).

💡Paired t-test

A paired t-test is a type of t-test used when the data consists of matched pairs or when the same subjects or entities are measured twice (e.g., before and after an intervention). In the video, the paired t-test is explained using the example of life expectancy in Africa in 1957 and 2007, where each country or observation in 1957 has a corresponding observation in 2007. The paired t-test accounts for the correlation between the pairs.

💡One-tailed and two-tailed tests

One-tailed and two-tailed tests refer to the direction of the alternative hypothesis in hypothesis testing. A one-tailed test is used when the alternative hypothesis specifies a direction (e.g., life expectancy in Africa is less than Europe), while a two-tailed test is used when the alternative hypothesis does not specify a direction (e.g., life expectancy in Ireland and Switzerland are different). The video explains and illustrates the use of one-tailed and two-tailed tests in the context of t-tests.

💡Confidence interval

A confidence interval is a range of values used to estimate a population parameter (e.g., mean, proportion) with a certain level of confidence. The video discusses confidence intervals in the context of ANOVA, where multiple comparison tests are performed to identify which group means differ. The confidence interval indicates the plausible range of values for the difference between two group means, and if it includes zero, it suggests the difference is not statistically significant.

💡Statistical significance

Statistical significance refers to the likelihood that an observed result or effect is not due to chance alone. It is determined by comparing the p-value (probability of obtaining the observed result if the null hypothesis is true) to a predetermined significance level (e.g., 0.05). If the p-value is less than the significance level, the result is considered statistically significant, and the null hypothesis is rejected. The video emphasizes the importance of statistical significance in interpreting the results of various statistical tests.

Highlights

When doing statistical tests, it's important to understand the question you're asking, and then it becomes easy to understand and interpret the results and decide which test to do.

For the t-test, there are single sample, two-tailed, one-tailed, and paired tests.

When doing a t-test, we're asking about the difference in means or averages between two populations, one population at different points in time, or between a sample mean and a presumed mean.

In hypothesis testing, we assume the null hypothesis (no difference in means), and if the probability of getting our sample data is very low under this assumption, we reject the null and accept that the difference is statistically significant.

For a single sample t-test, we compare the sample mean to a presumed population mean to see if the difference is statistically significant.

For a two-sample t-test, we can do a two-tailed test (is there a difference in any direction?) or a one-tailed test (is the difference in a particular direction?).

A paired t-test is used when there are pairs of observations, one from each population, matched together (e.g., life expectancy in Africa in 1957 and 2007).

For comparing means of three or more populations, we use ANOVA (analysis of variance).

After finding a significant difference with ANOVA, we can use multiple comparison tests to tease out which specific population means differ.

The chi-squared test is used for categorical variables and proportions, testing if the proportions across categories differ from expected values.

The chi-squared goodness of fit test checks if the observed proportions differ significantly from expected equal proportions.

The chi-squared test of independence checks if the proportions of one categorical variable depend on the values of another categorical variable.

In all tests, if the p-value is below a predetermined threshold (usually 0.05), we reject the null hypothesis and conclude the observed difference or relationship is statistically significant.

The key principles are the same across tests: assuming a null hypothesis of no difference, calculating the probability of getting the observed data under that assumption, and rejecting the null if that probability is very low.

Proper statistical testing requires determining the threshold for significance beforehand, to avoid p-hacking (adjusting criteria based on results).

Transcripts

welcome back to this global health

youtube channel in this video we're

going to be talking about statistical

tests which test to do when it's not

complicated and the key is to understand

what question it is that you're asking

and when you understand the question it

becomes very easy to understand and

interpret the results and to decide

which tests to do when so don't go away

stick with me you're going to love this

we're going to cover three things in

this video the t-test anova which is

analysis of variance and the chi-squared

test for the t-test there'll be the

single sample there's going to be

two-tailed one-tailed and paired right

we're going to do all four of those

things and for the chi squared there's

the goodness of fit and of course

there's also the test of independence

right you're going to find all of this

super duper easy to understand so stick

with me don't go away let's do this

let's talk about the t-test right and

i've got real data and real examples

here and we're going to look at four

different scenarios four different

applications of the t-test looking at

this data when we do a t-test we're

asking a question about the difference

in means or averages difference between

two populations or difference between

one population at different points in

time or the difference in an average

mean that we're seeing compared to some

sort of hypothesized or presumed average

right or presumed mean and we're asking

the question is our sample can from our

sample diet can we make inference about

the wider population is this somehow

representative of the truth that's out

there representative of the population

from which the sample was taken in other

words is this statistically significant

and this is how hypothesis testing works

we assume the opposite the

counterfactual the antithesis we assume

that there's no difference in means

between these two populations

if that were true and we call that the

null hypothesis if that were true

then how likely would it be that we

would get a sample

from the population that shows a

difference that we're seeing

or greater what is the probability of

that if we find that that is very

improbable and we decide up front by the

way we decide upfront what we consider

what the threshold for what we would

consider to be very improbable

what do we mean by very and we we often

use five percent if it's five percent or

less in terms of likelihood or

probability if we cons if it would be

very improbable to get a sample

like we've gotten

if the null hypothesis were true

and it would be very improbable to get

the sample but we have gotten the sample

we can then make the inference that the

null hypothesis must in fact be

incorrect that this assumption that

there's no difference that that's

incorrect we can reject that and we can

accept the fact that there is in fact a

difference and that our sample data is

statistically significant and that's how

hypothesis testing works so the first

example here at the top on the left is a

single sample t-test in other words

we've just got one sample just in this

case we've just got life expectancy in

africa so we've just taken africa we've

got life expectancy uh we don't have two

populations we've got one population

it's just africa and we've got a

presumed life expectancy or presumed

mean it could be for any old reason

there could be any reason why we believe

that life expectancy should be 50 or

should be 55 it could be any number and

we would ask the question is that the

sample that we've got

48.9 years is that statistically

significantly different from that

presumed mean or that presumed

population mean and then you get you

basically get a p-value if that p-value

is less than point zero five or five

percent if that if that's the number

that you've chosen as a threshold it

could be different

uh then if if it's if it's less than

that threshold and it's usually 0.00.05

then we can reject the null hypothesis

that the average is whatever it is that

we assumed it to be and we can accept

the fact that the the the the the sample

mean that we've gotten is statistically

significant okay that's the easiest

example if you understand that one

you'll understand the rest of these

super duper easy let's keep going in

these two examples and i've got two here

just to illustrate the fact that there's

two possible ways of asking this next

kind of question we've got we've got

life expectancy in africa and and in

europe in the top right hand corner and

we've got life expectancy in ireland and

in switzerland at the bottom bottom left

over here now the reason i've got two

here is just to highlight the fact that

there's two ways of doing this we could

ask the question

is there

a difference without specifying in which

direction so is there a difference for

example between the life expectancy in

ireland and the life expectancy in

switzerland and we might say look we

don't know in which direction the

difference might be we're just asking

are these are they the same or are they

different is the difference that we're

seeing here statistically significant

would we expect to see a difference of

this magnitude

or more if it were the case that in you

know in the scenario of the null

hypothesis that in fact ireland and

switzerland have the exact same

life expectancy and if and if that

probability is less than 0.05 if that's

our threshold then we would say that it

is statistically significant we reject

the null hypothesis we reject the notion

that they've got the same life

expectancy and we accept the fact that a

life expectancy that that this is

statistically significant we could

approach the same

the exact same problem in a slightly

different way and let's look let's use

africa and europe for that we could say

look we want to ask the question

is it statistically significant that

that africa is

that that africa has a life expectancy

less than europe of this magnitude so we

might go into the question saying we

know that africa has got a life

expectancy less than europe

we're asking

is a difference of this magnitude

statistically significant so we're not

asking is there a difference in any

direction

we're saying there is a we think there

is a difference

we think that the difference we think

that africa has a a different

we're asking is africa is the life

expectancy in africa less than europe

by this magnitude or more and is that

statistically significant and under

those circumstances you do a one-tailed

t-test does that make sense two-tailed

if you're saying is there a difference

in any direction one tailed if you're

saying is there a difference in a

particular direction

okay you got it and in both cases the

null hypothesis is that both populations

have got the same mean that that there's

no difference between the life

expectancy in the two populations now

here's the last example and this is this

is what i want you to understand this is

a paired tea test so we've got a one a

one tailed and a two-tailed over there a

paired t-test and this illustrates it

quite nicely

we've got a life expectancy in africa in

1957 and then the life expectancy in

africa in 2007.

so it's

for each sample

for each observation in

the 1957 there is a counterpart in 2007

right so

one in africa in 1957 one of the

examples would be south africa there'd

be a life expectancy that's one of the

observations but in 2007 there would

also be an observation that was south

africa right and there'd be an a life

expectancy there's malawi in both cases

zimbabwe in both in both in both samples

in other words there are these pairs

there is a counterpart in each

population and under those circumstances

you do a paired t test and all of the

principles the exact same principles

apply right you can it can be one-sided

two-sided

and and of course the p-value of less

than a threshold that you just determine

up front

anything less than that threshold if

it's if it's if it's five percent means

that the null hypothesis gets rejected

the null hypothesis is that both of

these have got the exact same life

expectancy the exact same mean and if

that's not the case if we reject that we

can accept that the difference that

we're seeing is

statistically significant all right got

it let's keep going next we're going to

talk about anova so here we've got

the we've got two means right if we want

to add a third what do we do we can't do

we can't do three means with the t-test

we would do an analysis of variance

anova all right so let's look at that

next

analysis of variance we're trying to

basically answer the same kind of

question that we were with the t-test

but except now we've got

three populations and we want to com

three or more populations and we wanted

to compare the means right the null

hypothesis is still the same that the

null hypothesis is that there are no

differences in the means of these

populations

and the alternative hypothesis is that

in fact there is a difference that the

difference we're seeing is statistically

significant right in this case i've used

box plots and density uh density plots

to to illustrate the difference in means

in three populations we've got europe

the americas and asia this is taken from

the gap-minded data this is real data so

it's super duper interesting and we're

seeing here that the data is showing a

difference in the means across these

different population groups right the

question is is that difference real and

if we do a an anova test it will show a

p-value that's very small but once

you've done that right you've done the

anova test you've shown that there's a

small p value we can reject the null

hypothesis which is that there's no

difference in the means we can accept

the fact that there is some sort of

statistical difference uh how then do we

tease out where that what's driving that

difference right because all that

conclusion all we can conclude from that

is that

one of

these populations is different from the

others and sometimes there's that you

know you may have more than three here

and you can do there are ways of

determining

uh

where that what's driving that

difference

and what i've done here is i've fed our

model into what's called a two key

multiple comparison of means and it's

taken each of the options you know asia

and america europe and america and

europe and asia and looked at them

individually right and if you look at

the results of it it's quite interesting

because you can sort of see well

between asia and america

there is a difference of

minus two or the difference of 2 it

doesn't really matter the magnitude

doesn't matter but the confidence

interval for that difference is between

-6 and 0.72 now and here you can see it

diagrammatically that confidence

interval crosses the zero threshold in

other words the confidence interval the

95 confidence interval includes the

possibility of a zero which a zero means

there's zero difference in other words

it includes the possibility that there's

no difference between those two

population means however europe and

america right we've got a difference of

four the confidence interval does not

include zero it's from zero point three

to seven point two

and uh and in europe to asia again same

story quite a big difference and the

difference does not include the

possibility of zero of no difference and

as you would expect the p values the

adjusted p values uh

bears that out so in the asia to america

where the confidence interval included

zero the p value does not cross the

threshold of of a five percent or less

it's 0.14 but the other two where the

confidence interval does not include

zero does not include the possibility of

no difference they both have small p

values less than point zero

point zero

point zero five okay got it

now let's talk about the chi-squared

test there's two of them right there's

the goodness of fatigue test and there's

the chi squared test of independence now

really what we're looking at here is

categorical variables and proportions of

categorical variables and this is a

great test to kind of test the notion

right we're testing whether or not there

in fact is a difference in the

proportions across the different

categories okay so let's have a look at

that right so here we've got

some flowers these happen to be irises

and we know that they come in we've

categorized them as small medium and

large

and in the first instance we could ask

the question are the proportion of

flowers that are small medium large the

same do we expect to see the same number

of small medium and large flowers in a

random sample that we take from the

population and we answer that question

by doing a chi-squared goodness of fit

test let's have a look at that

and again we're talking about hypothesis

testing in other words

if if it were the case

that there wasn't a difference in

proportions that would be our null

hypothesis right our null hypothesis is

that there's no difference in proportion

in the proportion of small medium and

large

flowers

right if that were true and we took a

random sample and that random sample

happened to show a difference in

proportions as large as or bigger than

the difference we're seeing now we would

consider if if that if if that

eventuality was considered to be

extremely unlikely then we could reject

the idea that they're all the same

proportions and we could accept the fact

that what we're seeing in the data is in

fact statistically significant what do

we mean by extremely unlikely well we've

got a cut-off point called the alpha

value that's the probability you know

how small this how small must that

probability be for us to consider it to

be

not like like

unacceptably smaller to the point that

we wouldn't really believe that we

wouldn't accept the null hypothesis to

be true

right and and usually we use 0.5 as

we've talked about so many times in

these videos

so we do a chi-square test we take the

starter we make a table put it into the

chi-square test and we get a p-value

that p-value is that probability if that

p-value is less than the threshold we

talked about usually 0.05 but it could

be anything depending on what it is that

you're trying to measure and how

important your sort of discrimination is

if that p-value is less than the

threshold and we reject the null we

accept the alternative and we say that

this difference that we're seeing in the

data is in fact statistically

significant and the exact same principle

applies to the chi-square test of

independence we're asking the question

are the proportions of

our species

are they in any way dependent or are

they independent of the size of the

flowers is knowing the value

you knowing this is does knowing the

size of the flower tell us anything

about the probability of a particular

flower being in one of these species

right looking at these graphs it seems

that that is the case but we need to

demonstrate that statistically so we do

a

chi-square test of independence and we

get a p-value if the p-value is very

very small beyond a pre-determined

threshold remember it has to be

predetermined you can't do it rest

retrospectively that's p hacking bad

science

uh if it's if the p value is very small

in other words the probability of a

sample showing a a difference in

proportions or a a relationship which is

demonstrated by difference in

proportions of this magnitude or more

the probability of that being the case

in the event that the null hypothesis

was was true in other words that there

was no difference if that proper

probability is extremely small we reject

the notion that these things are all the

same that the proportions are the same

and accept the fact that in fact what

we're seeing in the data this difference

that we're seeing this relationship that

we're seeing is in fact statistically

significant

right that's the chi-squared test of

independence now stay and watch another

video share this video with people that

you think might find it useful subscribe

to this channel if you haven't hit the

bell notification if you want

notification of future videos

take care

don't do drugs always do your best don't

ever change speak to you again soon