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Hypothesis Testing – A Deep Dive into Hypothesis Testing, The Backbone of Statistical Inference

• September 21, 2023

Explore the intricacies of hypothesis testing, a cornerstone of statistical analysis. Dive into methods, interpretations, and applications for making data-driven decisions.

In this Blog post we will learn:

• What is Hypothesis Testing?
• Steps in Hypothesis Testing 2.1. Set up Hypotheses: Null and Alternative 2.2. Choose a Significance Level (α) 2.3. Calculate a test statistic and P-Value 2.4. Make a Decision
• Example : Testing a new drug.
• Example in python

1. What is Hypothesis Testing?

In simple terms, hypothesis testing is a method used to make decisions or inferences about population parameters based on sample data. Imagine being handed a dice and asked if it’s biased. By rolling it a few times and analyzing the outcomes, you’d be engaging in the essence of hypothesis testing.

Think of hypothesis testing as the scientific method of the statistics world. Suppose you hear claims like “This new drug works wonders!” or “Our new website design boosts sales.” How do you know if these statements hold water? Enter hypothesis testing.

2. Steps in Hypothesis Testing

• Set up Hypotheses : Begin with a null hypothesis (H0) and an alternative hypothesis (Ha).
• Choose a Significance Level (α) : Typically 0.05, this is the probability of rejecting the null hypothesis when it’s actually true. Think of it as the chance of accusing an innocent person.
• Calculate Test statistic and P-Value : Gather evidence (data) and calculate a test statistic.
• p-value : This is the probability of observing the data, given that the null hypothesis is true. A small p-value (typically ≤ 0.05) suggests the data is inconsistent with the null hypothesis.
• Decision Rule : If the p-value is less than or equal to α, you reject the null hypothesis in favor of the alternative.

2.1. Set up Hypotheses: Null and Alternative

Before diving into testing, we must formulate hypotheses. The null hypothesis (H0) represents the default assumption, while the alternative hypothesis (H1) challenges it.

For instance, in drug testing, H0 : “The new drug is no better than the existing one,” H1 : “The new drug is superior .”

2.2. Choose a Significance Level (α)

When You collect and analyze data to test H0 and H1 hypotheses. Based on your analysis, you decide whether to reject the null hypothesis in favor of the alternative, or fail to reject / Accept the null hypothesis.

The significance level, often denoted by $α$, represents the probability of rejecting the null hypothesis when it is actually true.

In other words, it’s the risk you’re willing to take of making a Type I error (false positive).

Type I Error (False Positive) :

• Symbolized by the Greek letter alpha (α).
• Occurs when you incorrectly reject a true null hypothesis . In other words, you conclude that there is an effect or difference when, in reality, there isn’t.
• The probability of making a Type I error is denoted by the significance level of a test. Commonly, tests are conducted at the 0.05 significance level , which means there’s a 5% chance of making a Type I error .
• Commonly used significance levels are 0.01, 0.05, and 0.10, but the choice depends on the context of the study and the level of risk one is willing to accept.

Example : If a drug is not effective (truth), but a clinical trial incorrectly concludes that it is effective (based on the sample data), then a Type I error has occurred.

Type II Error (False Negative) :

• Symbolized by the Greek letter beta (β).
• Occurs when you accept a false null hypothesis . This means you conclude there is no effect or difference when, in reality, there is.
• The probability of making a Type II error is denoted by β. The power of a test (1 – β) represents the probability of correctly rejecting a false null hypothesis.

Example : If a drug is effective (truth), but a clinical trial incorrectly concludes that it is not effective (based on the sample data), then a Type II error has occurred.

Balancing the Errors :

In practice, there’s a trade-off between Type I and Type II errors. Reducing the risk of one typically increases the risk of the other. For example, if you want to decrease the probability of a Type I error (by setting a lower significance level), you might increase the probability of a Type II error unless you compensate by collecting more data or making other adjustments.

It’s essential to understand the consequences of both types of errors in any given context. In some situations, a Type I error might be more severe, while in others, a Type II error might be of greater concern. This understanding guides researchers in designing their experiments and choosing appropriate significance levels.

2.3. Calculate a test statistic and P-Value

Test statistic : A test statistic is a single number that helps us understand how far our sample data is from what we’d expect under a null hypothesis (a basic assumption we’re trying to test against). Generally, the larger the test statistic, the more evidence we have against our null hypothesis. It helps us decide whether the differences we observe in our data are due to random chance or if there’s an actual effect.

P-value : The P-value tells us how likely we would get our observed results (or something more extreme) if the null hypothesis were true. It’s a value between 0 and 1. – A smaller P-value (typically below 0.05) means that the observation is rare under the null hypothesis, so we might reject the null hypothesis. – A larger P-value suggests that what we observed could easily happen by random chance, so we might not reject the null hypothesis.

2.4. Make a Decision

Relationship between $α$ and P-Value

When conducting a hypothesis test:

We then calculate the p-value from our sample data and the test statistic.

Finally, we compare the p-value to our chosen $α$:

• If $p−value≤α$: We reject the null hypothesis in favor of the alternative hypothesis. The result is said to be statistically significant.
• If $p−value>α$: We fail to reject the null hypothesis. There isn’t enough statistical evidence to support the alternative hypothesis.

3. Example : Testing a new drug.

Imagine we are investigating whether a new drug is effective at treating headaches faster than drug B.

Setting Up the Experiment : You gather 100 people who suffer from headaches. Half of them (50 people) are given the new drug (let’s call this the ‘Drug Group’), and the other half are given a sugar pill, which doesn’t contain any medication.

• Set up Hypotheses : Before starting, you make a prediction:
• Null Hypothesis (H0): The new drug has no effect. Any difference in healing time between the two groups is just due to random chance.
• Alternative Hypothesis (H1): The new drug does have an effect. The difference in healing time between the two groups is significant and not just by chance.

Calculate Test statistic and P-Value : After the experiment, you analyze the data. The “test statistic” is a number that helps you understand the difference between the two groups in terms of standard units.

For instance, let’s say:

• The average healing time in the Drug Group is 2 hours.
• The average healing time in the Placebo Group is 3 hours.

The test statistic helps you understand how significant this 1-hour difference is. If the groups are large and the spread of healing times in each group is small, then this difference might be significant. But if there’s a huge variation in healing times, the 1-hour difference might not be so special.

Imagine the P-value as answering this question: “If the new drug had NO real effect, what’s the probability that I’d see a difference as extreme (or more extreme) as the one I found, just by random chance?”

For instance:

• P-value of 0.01 means there’s a 1% chance that the observed difference (or a more extreme difference) would occur if the drug had no effect. That’s pretty rare, so we might consider the drug effective.
• P-value of 0.5 means there’s a 50% chance you’d see this difference just by chance. That’s pretty high, so we might not be convinced the drug is doing much.
• If the P-value is less than ($α$) 0.05: the results are “statistically significant,” and they might reject the null hypothesis , believing the new drug has an effect.
• If the P-value is greater than ($α$) 0.05: the results are not statistically significant, and they don’t reject the null hypothesis , remaining unsure if the drug has a genuine effect.

4. Example in python

For simplicity, let’s say we’re using a t-test (common for comparing means). Let’s dive into Python:

Making a Decision : “The results are statistically significant! p-value < 0.05 , The drug seems to have an effect!” If not, we’d say, “Looks like the drug isn’t as miraculous as we thought.”

5. Conclusion

Hypothesis testing is an indispensable tool in data science, allowing us to make data-driven decisions with confidence. By understanding its principles, conducting tests properly, and considering real-world applications, you can harness the power of hypothesis testing to unlock valuable insights from your data.

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Best Guesses: Understanding The Hypothesis in Machine Learning

• February 22, 2024
• General , Supervised Learning , Unsupervised Learning

Machine learning is a vast and complex field that has inherited many terms from other places all over the mathematical domain.

It can sometimes be challenging to get your head around all the different terminologies, never mind trying to understand how everything comes together.

In this blog post, we will focus on one particular concept: the hypothesis.

While you may think this is simple, there is a little caveat regarding machine learning.

The statistics side and the learning side.

Don’t worry; we’ll do a full breakdown below.

You’ll learn the following:

What Is a Hypothesis in Machine Learning?

• Is This any different than the hypothesis in statistics?
• What is the difference between the alternative hypothesis and the null?
• Why do we restrict hypothesis space in artificial intelligence?
• Example code performing hypothesis testing in machine learning

In machine learning, the term ‘hypothesis’ can refer to two things.

First, it can refer to the hypothesis space, the set of all possible training examples that could be used to predict or answer a new instance.

Second, it can refer to the traditional null and alternative hypotheses from statistics.

Since machine learning works so closely with statistics, 90% of the time, when someone is referencing the hypothesis, they’re referencing hypothesis tests from statistics.

Is This Any Different Than The Hypothesis In Statistics?

In statistics, the hypothesis is an assumption made about a population parameter.

The statistician’s goal is to prove it true or disprove it.

This will take the form of two different hypotheses, one called the null, and one called the alternative.

Usually, you’ll establish your null hypothesis as an assumption that it equals some value.

For example, in Welch’s T-Test Of Unequal Variance, our null hypothesis is that the two means we are testing (population parameter) are equal.

This means our null hypothesis is that the two population means are the same.

We run our statistical tests, and if our p-value is significant (very low), we reject the null hypothesis.

This would mean that their population means are unequal for the two samples you are testing.

Usually, statisticians will use the significance level of .05 (a 5% risk of being wrong) when deciding what to use as the p-value cut-off.

What Is The Difference Between The Alternative Hypothesis And The Null?

The null hypothesis is our default assumption, which we are trying to prove correct.

The alternate hypothesis is usually the opposite of our null and is much broader in scope.

For most statistical tests, the null and alternative hypotheses are already defined.

You are then just trying to find “significant” evidence we can use to reject our null hypothesis.

These two hypotheses are easy to spot by their specific notation. The null hypothesis is usually denoted by H₀, while H₁ denotes the alternative hypothesis.

Example Code Performing Hypothesis Testing In Machine Learning

Since there are many different hypothesis tests in machine learning and data science, we will focus on one of my favorites.

This test is Welch’s T-Test Of Unequal Variance, where we are trying to determine if the population means of these two samples are different.

There are a couple of assumptions for this test, but we will ignore those for now and show the code.

You can read more about this here in our other post, Welch’s T-Test of Unequal Variance .

We see that our p-value is very low, and we reject the null hypothesis.

What Is The Difference Between The Biased And Unbiased Hypothesis Spaces?

The difference between the Biased and Unbiased hypothesis space is the number of possible training examples your algorithm has to predict.

The unbiased space has all of them, and the biased space only has the training examples you’ve supplied.

Since neither of these is optimal (one is too small, one is much too big), your algorithm creates generalized rules (inductive learning) to be able to handle examples it hasn’t seen before.

Here’s an example of each:

Example of The Biased Hypothesis Space In Machine Learning

The Biased Hypothesis space in machine learning is a biased subspace where your algorithm does not consider all training examples to make predictions.

This is easiest to see with an example.

Let’s say you have the following data:

Happy  and  Sunny  and  Stomach Full  = True

Whenever your algorithm sees those three together in the biased hypothesis space, it’ll automatically default to true.

This means when your algorithm sees:

Sad  and  Sunny  And  Stomach Full  = False

It’ll automatically default to False since it didn’t appear in our subspace.

This is a greedy approach, but it has some practical applications.

Example of the Unbiased Hypothesis Space In Machine Learning

The unbiased hypothesis space is a space where all combinations are stored.

We can use re-use our example above:

This would start to breakdown as

Happy  = True

Happy  and  Sunny  = True

Happy  and  Stomach Full  = True

Let’s say you have four options for each of the three choices.

This would mean our subspace would need 2^12 instances (4096) just for our little three-word problem.

This is practically impossible; the space would become huge.

So while it would be highly accurate, this has no scalability.

More reading on this idea can be found in our post, Inductive Bias In Machine Learning .

Why Do We Restrict Hypothesis Space In Artificial Intelligence?

We have to restrict the hypothesis space in machine learning. Without any restrictions, our domain becomes much too large, and we lose any form of scalability.

This is why our algorithm creates rules to handle examples that are seen in production. 

This gives our algorithms a generalized approach that will be able to handle all new examples that are in the same format.

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Below we’ve listed a few that are similar to this guide:

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What is hypothesis in Machine Learning?

The hypothesis is a word that is frequently used in Machine Learning and data science initiatives. As we all know, machine learning is one of the most powerful technologies in the world, allowing us to anticipate outcomes based on previous experiences. Moreover, data scientists and ML specialists undertake experiments with the goal of solving an issue. These ML experts and data scientists make an initial guess on how to solve the challenge.

What is a Hypothesis?

A hypothesis is a conjecture or proposed explanation that is based on insufficient facts or assumptions. It is only a conjecture based on certain known facts that have yet to be confirmed. A good hypothesis is tested and yields either true or erroneous outcomes.

Let's look at an example to better grasp the hypothesis. According to some scientists, ultraviolet (UV) light can harm the eyes and induce blindness.

In this case, a scientist just states that UV rays are hazardous to the eyes, but people presume they can lead to blindness. Yet, it is conceivable that it will not be achievable. As a result, these kinds of assumptions are referred to as hypotheses.

Defining Hypothesis in Machine Learning

In machine learning, a hypothesis is a mathematical function or model that converts input data into output predictions. The model's first belief or explanation is based on the facts supplied. The hypothesis is typically expressed as a collection of parameters characterizing the behavior of the model.

If we're building a model to predict the price of a property based on its size and location. The hypothesis function may look something like this −

$$\mathrm{h(x)\:=\:θ0\:+\:θ1\:*\:x1\:+\:θ2\:*\:x2}$$

The hypothesis function is h(x), its input data is x, the model's parameters are 0, 1, and 2, and the features are x1 and x2.

The machine learning model's purpose is to discover the optimal values for parameters 0 through 2 that minimize the difference between projected and actual output labels.

To put it another way, we're looking for the hypothesis function that best represents the underlying link between the input and output data.

Types of Hypotheses in Machine Learning

The next step is to build a hypothesis after identifying the problem and obtaining evidence. A hypothesis is an explanation or solution to a problem based on insufficient data. It acts as a springboard for further investigation and experimentation. A hypothesis is a machine learning function that converts inputs to outputs based on some assumptions. A good hypothesis contributes to the creation of an accurate and efficient machine-learning model. Several machine learning theories are as follows −

1. Null Hypothesis

A null hypothesis is a basic hypothesis that states that no link exists between the independent and dependent variables. In other words, it assumes the independent variable has no influence on the dependent variable. It is symbolized by the symbol H0. If the p-value falls outside the significance level, the null hypothesis is typically rejected (). If the null hypothesis is correct, the coefficient of determination is the probability of rejecting it. A null hypothesis is involved in test findings such as t-tests and ANOVA.

2. Alternative Hypothesis

An alternative hypothesis is a hypothesis that contradicts the null hypothesis. It assumes that there is a relationship between the independent and dependent variables. In other words, it assumes that there is an effect of the independent variable on the dependent variable. It is denoted by Ha. An alternative hypothesis is generally accepted if the p-value is less than the significance level (α). An alternative hypothesis is also known as a research hypothesis.

3. One-tailed Hypothesis

A one-tailed test is a type of significance test in which the region of rejection is located at one end of the sample distribution. It denotes that the estimated test parameter is more or less than the crucial value, implying that the alternative hypothesis rather than the null hypothesis should be accepted. It is most commonly used in the chi-square distribution, where all of the crucial areas, related to, are put in either of the two tails. Left-tailed or right-tailed one-tailed tests are both possible.

4. Two-tailed Hypothesis

The two-tailed test is a hypothesis test in which the region of rejection or critical area is on both ends of the normal distribution. It determines whether the sample tested falls within or outside a certain range of values, and an alternative hypothesis is accepted if the calculated value falls in either of the two tails of the probability distribution. α is bifurcated into two equal parts, and the estimated parameter is either above or below the assumed parameter, so extreme values work as evidence against the null hypothesis.

Overall, the hypothesis plays a critical role in the machine learning model. It provides a starting point for the model to make predictions and helps to guide the learning process. The accuracy of the hypothesis is evaluated using various metrics like mean squared error or accuracy.

The hypothesis is a mathematical function or model that converts input data into output predictions, typically expressed as a collection of parameters characterizing the behavior of the model. It is an explanation or solution to a problem based on insufficient data. A good hypothesis contributes to the creation of an accurate and efficient machine-learning model. A two-tailed hypothesis is used when there is no prior knowledge or theoretical basis to infer a certain direction of the link.

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Hypothesis in Machine Learning: Comprehensive Overview(2021)

Introduction

Supervised machine learning (ML) is regularly portrayed as the issue of approximating an objective capacity that maps inputs to outputs. This portrayal is described as looking through and assessing competitor hypothesis from hypothesis spaces. 

The conversation of hypothesis in machine learning can be confused for a novice, particularly when “hypothesis” has a discrete, but correlated significance in statistics and all the more comprehensively in science.

Hypothesis Space (H)

The hypothesis space utilized by an ML system is the arrangement of all hypotheses that may be returned by it. It is ordinarily characterized by a Hypothesis Language, conceivably related to a Language Bias. 

Many ML algorithms depend on some sort of search methodology: given a set of perceptions and a space of all potential hypotheses that may be thought in the hypothesis space. They see in this space for those hypotheses that adequately furnish the data or are ideal concerning some other quality standard.

ML can be portrayed as the need to utilize accessible data objects to discover a function that most reliable maps inputs to output, alluded to as function estimate, where we surmised an anonymous objective function that can most reliably map inputs to outputs on all expected perceptions from the difficult domain. An illustration of a model that approximates the performs mappings and target function of inputs to outputs is known as hypothesis testing in machine learning.

The hypothesis in machine learning of all potential hypothesis that you are looking over, paying little mind to their structure. For the wellbeing of accommodation, the hypothesis class is normally compelled to be just each sort of function or model in turn, since learning techniques regularly just work on each type at a time. This doesn’t need to be the situation, however:

• Hypothesis classes don’t need to comprise just one kind of function. If you’re looking through exponential, quadratic, and overall linear functions, those are what your joined hypothesis class contains.
• Hypothesis classes additionally don’t need to comprise of just straightforward functions. If you figure out how to look over all piecewise-tanh2 functions, those functions are what your hypothesis class incorporates.

The enormous trade-off is that the bigger your hypothesis class in   machine learning, the better the best hypothesis models the basic genuine function, yet the harder it is to locate that best hypothesis. This is identified with the bias-variance trade-off.

• Hypothesis (h)

A hypothesis function in machine learning is best describes the target. The hypothesis that an algorithm would concoct relies on the data and relies on the bias and restrictions that we have forced on the data.

The hypothesis formula in machine learning:

• y  is range
• m  changes in y divided by change in x
• x  is domain
• b  is intercept

The purpose of restricting hypothesis space in machine learning is so that these can fit well with the general data that is needed by the user. It checks the reality or deception of observations or inputs and examinations them appropriately. Subsequently, it is extremely helpful and it plays out the valuable function of mapping all the inputs till they come out as outputs. Consequently, the target functions are deliberately examined and restricted dependent on the outcomes (regardless of whether they are free of bias), in ML.

The hypothesis in machine learning space and inductive bias in machine learning is that the hypothesis space is a collection of valid Hypothesis, for example, every single desirable function, on the opposite side the inductive bias (otherwise called learning bias) of a learning algorithm is the series of expectations that the learner uses to foresee outputs of given sources of inputs that it has not experienced. Regression and Classification are a kind of realizing which relies upon continuous-valued and discrete-valued sequentially. This sort of issues (learnings) is called inductive learning issues since we distinguish a function by inducting it on data.

In the Maximum a Posteriori or MAP hypothesis in machine learning, enhancement gives a Bayesian probability structure to fitting model parameters to training data and another option and sibling may be a more normal Maximum Likelihood Estimation system. MAP learning chooses a solitary in all probability theory given the data. The hypothesis in machine learning earlier is as yet utilized and the technique is regularly more manageable than full Bayesian learning. 

Bayesian techniques can be utilized to decide the most plausible hypothesis in machine learning given the data the MAP hypothesis. This is the ideal hypothesis as no other hypothesis is more probable.

Hypothesis in machine learning or ML the applicant model that approximates a target function for mapping instances of inputs to outputs.

Hypothesis in statistics probabilistic clarification about the presence of a connection between observations. 

Hypothesis in science is a temporary clarification that fits the proof and can be disproved or confirmed. We can see that a hypothesis in machine learning draws upon the meaning of the hypothesis all the more extensively in science.

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Manifold Hypothesis

What is the Manifold Hypothesis?

The Manifold Hypothesis states that real-world high-dimensional data lie on low-dimensional manifolds embedded within the high-dimensional space.

This hypothesis is better explained in examples, however.

Let's tackle the "embedded manifold" bit first, before we get to how it applies to machine learning and data.

A manifold is really just a technical term that is used to classify spaces of arbitrary dimension. For every whole number there exists a flat space called Euclidean space that has characteristics very similar to the cartesian plane. For example the Pythagorean theorem holds and thus the shortest distance between points is a straight line (in contrast, this is not true on a circle or sphere).  The dimension of a Euclidean space is essentially the number of (independent) degrees of freedom - basically, the number of (orthogonal) directions one can "move" inside the space).  A line has one such dimension, an infinite plane has 2, and an  infinite volume has 3, and so n.  A manifold is essentially a generalization of Euclidean space such that locally (small areas) are approximately the same as Euclidean space but the entire space fails to be have the same properties of Euclidean space when observed in its entirety.  This theoretical framework always mathematicians and other quantitively motivated scientists to describe spaces, like spheres, tori (donut-shaped spaces) and mobius bands, in a precise way and even allows a whole plethora of mathematical machinery, including calculus, to be used in a meaningful way.  The upshot is that now the classes of spaces upon which calculus now makes sense is expanded to include spaces that may be curved in arbitrary ways, or even have holes like the torus. 

So now we take this idea, and apply it to high-dimensional data.   Imagine we are interested in classify all (black and white) mages with mxn pixels.  Each pixel has a numerical value, and each can vary depending on what the image is, which could correspond to anything from an award wining photo to meaningless noise.  The point is that we have mxn degrees of freedom so we can treat an image of mxn pixels as being a single point in living in a space (manifold)  of dimension N = mn ,  Now, imagine the set of all mxn imagines that are photos of Einstein.  Clearly we now have some restriction on the choice of values for the pixels if we want the images to be photos of Einstein rather than something else.  Obviously random choices will not generate such images.  Therefore, we expect there to be less freedom of choice and hence:

The manifold hypothesis states that that this subset should actually live in an (ambient) space of lower dimension, in fact a dimension much, much smaller than  N . 

Why This Hypothesis is Important in Artificial Intelligence? 

The Manifold Hypothesis explains ( heuristically ) why machine learning techniques are able to find useful features and produce accurate predictions from datasets that have a potentially large number of dimensions ( variables).    The fact that the actual data set of interest actually lives on in a space of low dimension, means that a given machine learning model only needs to learn to focus on a few key features of the dataset to make decisions.  However these key features may turn out to be complicated functions of the original variables.  Many of the algorithms behind machine learning techniques focus on ways to determine these (embedding) functions.

MIT has an excellent paper  on testing the hypothesis. We also recommend checking out  Colah’s blog .

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Hypothesis testing involves formulating assumptions about population parameters based on sample statistics and rigorously evaluating these assumptions against empirical evidence. This article sheds light on the significance of hypothesis testing and the critical steps involved in the process.

What is Hypothesis Testing?

Hypothesis testing is a statistical method that is used to make a statistical decision using experimental data. Hypothesis testing is basically an assumption that we make about a population parameter. It evaluates two mutually exclusive statements about a population to determine which statement is best supported by the sample data. 

Example: You say an average height in the class is 30 or a boy is taller than a girl. All of these is an assumption that we are assuming, and we need some statistical way to prove these. We need some mathematical conclusion whatever we are assuming is true.

Defining Hypotheses

Key Terms of Hypothesis Testing

• P-value: The P value , or calculated probability, is the probability of finding the observed/extreme results when the null hypothesis(H0) of a study-given problem is true. If your P-value is less than the chosen significance level then you reject the null hypothesis i.e. accept that your sample claims to support the alternative hypothesis.
• Test Statistic: The test statistic is a numerical value calculated from sample data during a hypothesis test, used to determine whether to reject the null hypothesis. It is compared to a critical value or p-value to make decisions about the statistical significance of the observed results.
• Critical value : The critical value in statistics is a threshold or cutoff point used to determine whether to reject the null hypothesis in a hypothesis test.
• Degrees of freedom: Degrees of freedom are associated with the variability or freedom one has in estimating a parameter. The degrees of freedom are related to the sample size and determine the shape.

Why do we use Hypothesis Testing?

Hypothesis testing is an important procedure in statistics. Hypothesis testing evaluates two mutually exclusive population statements to determine which statement is most supported by sample data. When we say that the findings are statistically significant, thanks to hypothesis testing. 

One-Tailed and Two-Tailed Test

One tailed test focuses on one direction, either greater than or less than a specified value. We use a one-tailed test when there is a clear directional expectation based on prior knowledge or theory. The critical region is located on only one side of the distribution curve. If the sample falls into this critical region, the null hypothesis is rejected in favor of the alternative hypothesis.

One-Tailed Test

There are two types of one-tailed test:

Two-Tailed Test

A two-tailed test considers both directions, greater than and less than a specified value.We use a two-tailed test when there is no specific directional expectation, and want to detect any significant difference.

What are Type 1 and Type 2 errors in Hypothesis Testing?

In hypothesis testing, Type I and Type II errors are two possible errors that researchers can make when drawing conclusions about a population based on a sample of data. These errors are associated with the decisions made regarding the null hypothesis and the alternative hypothesis.

How does Hypothesis Testing work?

Step 1: define null and alternative hypothesis.

We first identify the problem about which we want to make an assumption keeping in mind that our assumption should be contradictory to one another, assuming Normally distributed data.

Step 2 – Choose significance level

Step 3 – Collect and Analyze data.

Gather relevant data through observation or experimentation. Analyze the data using appropriate statistical methods to obtain a test statistic.

Step 4-Calculate Test Statistic

The data for the tests are evaluated in this step we look for various scores based on the characteristics of data. The choice of the test statistic depends on the type of hypothesis test being conducted.

There are various hypothesis tests, each appropriate for various goal to calculate our test. This could be a Z-test , Chi-square , T-test , and so on.

• Z-test : If population means and standard deviations are known. Z-statistic is commonly used.
• t-test : If population standard deviations are unknown. and sample size is small than t-test statistic is more appropriate.
• Chi-square test : Chi-square test is used for categorical data or for testing independence in contingency tables
• F-test : F-test is often used in analysis of variance (ANOVA) to compare variances or test the equality of means across multiple groups.

We have a smaller dataset, So, T-test is more appropriate to test our hypothesis.

T-statistic is a measure of the difference between the means of two groups relative to the variability within each group. It is calculated as the difference between the sample means divided by the standard error of the difference. It is also known as the t-value or t-score.

Step 5 – Comparing Test Statistic:

In this stage, we decide where we should accept the null hypothesis or reject the null hypothesis. There are two ways to decide where we should accept or reject the null hypothesis.

Method A: Using Crtical values

Comparing the test statistic and tabulated critical value we have,

• If Test Statistic>Critical Value: Reject the null hypothesis.
• If Test Statistic≤Critical Value: Fail to reject the null hypothesis.

Note: Critical values are predetermined threshold values that are used to make a decision in hypothesis testing. To determine critical values for hypothesis testing, we typically refer to a statistical distribution table , such as the normal distribution or t-distribution tables based on.

Method B: Using P-values

We can also come to an conclusion using the p-value,

Note : The p-value is the probability of obtaining a test statistic as extreme as, or more extreme than, the one observed in the sample, assuming the null hypothesis is true. To determine p-value for hypothesis testing, we typically refer to a statistical distribution table , such as the normal distribution or t-distribution tables based on.

Step 7- Interpret the Results

At last, we can conclude our experiment using method A or B.

Calculating test statistic

To validate our hypothesis about a population parameter we use statistical functions . We use the z-score, p-value, and level of significance(alpha) to make evidence for our hypothesis for normally distributed data .

1. Z-statistics:

When population means and standard deviations are known.

• μ represents the population mean, 
• σ is the standard deviation
• and n is the size of the sample.

2. T-Statistics

T test is used when n<30,

t-statistic calculation is given by:

• t = t-score,
• x̄ = sample mean
• μ = population mean,
• s = standard deviation of the sample,
• n = sample size

3. Chi-Square Test

Chi-Square Test for Independence categorical Data (Non-normally distributed) using:

• i,j are the rows and columns index respectively.

Real life Hypothesis Testing example

Let’s examine hypothesis testing using two real life situations,

Case A: D oes a New Drug Affect Blood Pressure?

Imagine a pharmaceutical company has developed a new drug that they believe can effectively lower blood pressure in patients with hypertension. Before bringing the drug to market, they need to conduct a study to assess its impact on blood pressure.

• Before Treatment: 120, 122, 118, 130, 125, 128, 115, 121, 123, 119
• After Treatment: 115, 120, 112, 128, 122, 125, 110, 117, 119, 114

Step 1 : Define the Hypothesis

• Null Hypothesis : (H 0 )The new drug has no effect on blood pressure.
• Alternate Hypothesis : (H 1 )The new drug has an effect on blood pressure.

Step 2: Define the Significance level

Let’s consider the Significance level at 0.05, indicating rejection of the null hypothesis.

If the evidence suggests less than a 5% chance of observing the results due to random variation.

Step 3 : Compute the test statistic

Using paired T-test analyze the data to obtain a test statistic and a p-value.

The test statistic (e.g., T-statistic) is calculated based on the differences between blood pressure measurements before and after treatment.

t = m/(s/√n)

• m  = mean of the difference i.e X after, X before
• s  = standard deviation of the difference (d) i.e d i ​= X after, i ​− X before,
• n  = sample size,

then, m= -3.9, s= 1.8 and n= 10

we, calculate the , T-statistic = -9 based on the formula for paired t test

Step 4: Find the p-value

The calculated t-statistic is -9 and degrees of freedom df = 9, you can find the p-value using statistical software or a t-distribution table.

thus, p-value = 8.538051223166285e-06

Step 5: Result

• If the p-value is less than or equal to 0.05, the researchers reject the null hypothesis.
• If the p-value is greater than 0.05, they fail to reject the null hypothesis.

Conclusion: Since the p-value (8.538051223166285e-06) is less than the significance level (0.05), the researchers reject the null hypothesis. There is statistically significant evidence that the average blood pressure before and after treatment with the new drug is different.

Python Implementation of Hypothesis Testing

Let’s create hypothesis testing with python, where we are testing whether a new drug affects blood pressure. For this example, we will use a paired T-test. We’ll use the scipy.stats library for the T-test.

Scipy is a mathematical library in Python that is mostly used for mathematical equations and computations.

We will implement our first real life problem via python,

In the above example, given the T-statistic of approximately -9 and an extremely small p-value, the results indicate a strong case to reject the null hypothesis at a significance level of 0.05. 

• The results suggest that the new drug, treatment, or intervention has a significant effect on lowering blood pressure.
• The negative T-statistic indicates that the mean blood pressure after treatment is significantly lower than the assumed population mean before treatment.

Case B : Cholesterol level in a population

Data: A sample of 25 individuals is taken, and their cholesterol levels are measured.

Cholesterol Levels (mg/dL): 205, 198, 210, 190, 215, 205, 200, 192, 198, 205, 198, 202, 208, 200, 205, 198, 205, 210, 192, 205, 198, 205, 210, 192, 205.

Populations Mean = 200

Population Standard Deviation (σ): 5 mg/dL(given for this problem)

Step 1: Define the Hypothesis

• Null Hypothesis (H 0 ): The average cholesterol level in a population is 200 mg/dL.
• Alternate Hypothesis (H 1 ): The average cholesterol level in a population is different from 200 mg/dL.

As the direction of deviation is not given , we assume a two-tailed test, and based on a normal distribution table, the critical values for a significance level of 0.05 (two-tailed) can be calculated through the z-table and are approximately -1.96 and 1.96.

Step 4: Result

Since the absolute value of the test statistic (2.04) is greater than the critical value (1.96), we reject the null hypothesis. And conclude that, there is statistically significant evidence that the average cholesterol level in the population is different from 200 mg/dL

Limitations of Hypothesis Testing

• Although a useful technique, hypothesis testing does not offer a comprehensive grasp of the topic being studied. Without fully reflecting the intricacy or whole context of the phenomena, it concentrates on certain hypotheses and statistical significance.
• The accuracy of hypothesis testing results is contingent on the quality of available data and the appropriateness of statistical methods used. Inaccurate data or poorly formulated hypotheses can lead to incorrect conclusions.
• Relying solely on hypothesis testing may cause analysts to overlook significant patterns or relationships in the data that are not captured by the specific hypotheses being tested. This limitation underscores the importance of complimenting hypothesis testing with other analytical approaches.

Hypothesis testing stands as a cornerstone in statistical analysis, enabling data scientists to navigate uncertainties and draw credible inferences from sample data. By systematically defining null and alternative hypotheses, choosing significance levels, and leveraging statistical tests, researchers can assess the validity of their assumptions. The article also elucidates the critical distinction between Type I and Type II errors, providing a comprehensive understanding of the nuanced decision-making process inherent in hypothesis testing. The real-life example of testing a new drug’s effect on blood pressure using a paired T-test showcases the practical application of these principles, underscoring the importance of statistical rigor in data-driven decision-making.

Frequently Asked Questions (FAQs)

1. what are the 3 types of hypothesis test.

There are three types of hypothesis tests: right-tailed, left-tailed, and two-tailed. Right-tailed tests assess if a parameter is greater, left-tailed if lesser. Two-tailed tests check for non-directional differences, greater or lesser.

2.What are the 4 components of hypothesis testing?

Null Hypothesis ( ): No effect or difference exists. Alternative Hypothesis ( ): An effect or difference exists. Significance Level ( ): Risk of rejecting null hypothesis when it’s true (Type I error). Test Statistic: Numerical value representing observed evidence against null hypothesis.

3.What is hypothesis testing in ML?

Statistical method to evaluate the performance and validity of machine learning models. Tests specific hypotheses about model behavior, like whether features influence predictions or if a model generalizes well to unseen data.

4.What is the difference between Pytest and hypothesis in Python?

Pytest purposes general testing framework for Python code while Hypothesis is a Property-based testing framework for Python, focusing on generating test cases based on specified properties of the code.

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In this video, Christopher Brooks, Associate Professor of Information, discusses the fundamentals of machine learning, including supervised learning (classification, regression), unsupervised learning (clustering), semisupervised learning, and reinforcement learning.

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Introduction to machine learning in sports analytics.

So what is machine learning? You can think of machine learning as a paradigm of building computational models based on historical data without you having to explicitly create rules about that data. We build these models in an iterative fashion. So we collect some data about a phenomenon. Maybe we watch a bunch of sports matches or athletic events. We then use statistical methods and we apply it to the data and the statistical methods organized, they find patterns inside of the data. And that creates a model which we can then use for prediction on new data. And you've already seen machine learning at work in this specialization through regression. So I want to break down the main branches of machine learning and they differ depending on the task. So supervised learning. Now this class is actually going to focus really on supervised learning. So the task with supervised learning is to learn the relationship between this historical data and some labels which exist now. The labels are usually provided by humans, and the goal of the models that we create is to predict labels for new data, which we don't have the label for. And there's really two broad categories of supervised approaches. In regression approaches, the label is a target value. So this might be the draft pick position of a player, which you might predict from their previous performance. In classification, the label is actually a class value, so this is categorical in nature. So this is like predicting the kind of activity that you might get out of sensor data or predicting who's going to win given match. Unsupervised learning approaches on the other hand, don't require this label. They take the historical data and they use this to identify the features from the data, which could be used to help us understand new data. Really the most common task is clustering of this data, and unsupervised approaches really tell us a relationship between the different observations inside of the data. And this is sometimes done just statistically, so we're interested in understanding what kind of clusters exist and what a centroid looks like. So does a good forward player look like or what does a good goalie look like? But sometimes we do this for disambiguation and we do this with visualization in mind. And so we actually want to visually look at people and their relationship and their activities inside of sports events to see if there's different patterns. So, in sports analytics, there's actually numerous great examples which players are similar based on their stats, which physical activity share similar sensor data, which teams have similar playing patterns. So despite not having a label, human decision making is still an important part of the process and this is really in determining what features we're going to feed into the model and that we're going to cluster on. So for instance, if you're clustering NHL players, if you fed in features such as goal scoring and location on ice, you're going to differentiate the players based on position. So forwards tend to score goals, and they tend to be all over the ice. A goalie tends not to score goals, although they have and tends to stay on one place on the ice. But if instead you start feeding in features such as time on ice, salary and so forth, the models coming back are going to tell you more about perceived value of players. So there might be a well paid forward, a well paid goalie, and a well paid defenseman all coming up in one cluster. Semi supervised learning. So the title here sort of gives it away. Semi supervised learning involves a mixture of supervised and unsupervised learning approaches where the human labelling of the data is expensive or incomplete. For instance, imagine that you've scraped the web and you've got a collection of thousands of pictures of athletes and you want to analyze what pair of shoes they're wearing using computer vision and machine learning. So individual shoe identification is actually a pretty difficult problem and it's quite error prone. And so what we might want to do instead is see the system with a few classifications, show an expert a few of the pictures and say. Yeah, those are this brand or they have this kind of logo on them or their this kind of trainer. And then the machine learning algorithms going to identify features of the images, color or logo features, vector features of what the logo looks like, or lacing or something like this. Then clusters a large set of the images into different clusters and it'll have some goodness of fit for each of those clusters. And so it'll be able to identify those pieces that seem to be outliers and then you can raise those up to humans and the humans can provide labels for those. Humans can also go to the clusters where you think you have it well thought out and spot check and provide new training data, thus adding the labels and then you repeat the process. Reinforcement learning. So reinforcement learning is actually really exciting to me. I really think that this is a very exciting technique. The goal in reinforcement learning is you're going to train using a supervised method but the human doesn't actually provide the labels. Instead the machine itself in this case, I mean the machine learning algorithm can sense the environment and can actually see the labels in the environment. So most commonly this is done by providing some reward function. That function rewards a machine for correctly classifying data in real time without human intervention. So an example of this in sports analytics might be in amateur athlete training where some broad objective is known and could be measured. For instance, let's say there's a rehabilitation program and you want compliance with a training program because an athlete has been injured and you can get compliance from wearable data. Did they go out and walk? Did they go out and run? Are they doing the activities that were prescribed to them? Let's say, by a physiotherapist or a trainer. Then the machine is able to take some interventions and try and make this happen. Try and change what the user activity is in the environment. So, for instance, sending an email or phone app nudges a little notification or a little reward, and so forth. And we experience these all the time right now. And some of these are based on machine learning. So the relationship then between when and how often to send these emails or other interventions, we start to learn by watching the effect they have on the compliance with the training program. So for instance, if emails don't seem to be working, maybe the machine has some other opportunities. Such as sending the nudges or pinging a physiotherapist and getting them involved to bring the player back into compliance. So this is very similar to a B or randomized controlled trial tests with the goal of trying to do some experimentation. But this is done with machine learning instead. And so we don't use equal proportions of subjects and we just start to learn which approaches are best, and then we start to heavily favor those. So in this brief lecture we talked about the machine learning space and there's really four main approaches to applying machine learning. Supervised approaches, like classification and regression are probably the most common and well explored in many, many domains, including sports analytics, and that's where we're going to focus our discussions in this class. Unsupervised approaches, though like clustering are also heavily used, especially when you're trying to understand some relationships between players based on play style. Semi supervised approaches are really an interesting blend of these two. And reinforcement learning is an excellent way to start experimenting and understanding and changing behaviors when your machine or your machine learning method can actually start to sense the environment.

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Abstract: Advances in computational power and AI have increased interest in reinforcement learning approaches to inventory management. This paper provides a theoretical foundation for these approaches and investigates the benefits of restricting to policy structures that are well-established by decades of inventory theory. In particular, we prove generalization guarantees for learning several well-known classes of inventory policies, including base-stock and (s, S) policies, by leveraging the celebrated Vapnik-Chervonenkis (VC) theory. We apply the concepts of the Pseudo-dimension and Fat-shattering dimension from VC theory to determine the generalizability of inventory policies, that is, the difference between an inventory policy's performance on training data and its expected performance on unseen data. We focus on a classical setting without contexts, but allow for an arbitrary distribution over demand sequences and do not make any assumptions such as independence over time. We corroborate our supervised learning results using numerical simulations. Managerially, our theory and simulations translate to the following insights. First, there is a principle of "learning less is more" in inventory management: depending on the amount of data available, it may be beneficial to restrict oneself to a simpler, albeit suboptimal, class of inventory policies to minimize overfitting errors. Second, the number of parameters in a policy class may not be the correct measure of overfitting error: in fact, the class of policies defined by T time-varying base-stock levels exhibits a generalization error comparable to that of the two-parameter (s, S) policy class. Finally, our research suggests situations in which it could be beneficial to incorporate the concepts of base-stock and inventory position into black-box learning machines, instead of having these machines directly learn the order quantity actions.

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Identification and Fault Diagnosis of Rolling Element Bearings Using Dimension Theory and Machine Learning Techniques

Contributed by the Tribology Division of ASME for publication in the J ournal of T ribology .

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Jadhav, P. S., Salunkhe, V. G., Desavale, R. G., Khot, S., Shinde, P. V., Jadhav, P. M., and Gadyanavar, P. R. (April 17, 2024). "Identification and Fault Diagnosis of Rolling Element Bearings Using Dimension Theory and Machine Learning Techniques." ASME. J. Tribol . doi: https://doi.org/10.1115/1.4065335

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The study presents the classification of bearing fault types occurring in rotating machines using machine learning techniques. Recent condition monitoring demands all-inclusive but precise fault diagnosis for industrial machines. Synchronizing mathematical modelling with machine learning may combine both potentials to enhance diagnosis efficiency. The current study presents a blend of dimensional analysis (DA) and a K- nearest neighbor (KNN) to diagnose faults in industrial roller bearings. Vibrational responses are collected for several industrial machines under diverse operational conditions. Bearing faults are identified using the DA model with 3.62 % error (avg) and classified using KNN with 98.67% accuracy. Comparing the performance of models with experimental and Artificial Neural Networks (ANN) validated the potential of the current approach. The results showed that the KNN demonstrates superior performance in terms of feature prediction and extraction of industrial bearing.

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RSC Advances

A prediction model for co 2 /co adsorption performance on binary alloys based on machine learning †.

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a School of Chemistry and Chemical Engineering, Southwest Petroleum University, Chengdu, P. R. China E-mail: [email protected]

Despite the rapid development of computational methods, including density functional theory (DFT), predicting the performance of a catalytic material merely based on its atomic arrangements remains challenging. Although quantum mechanics-based methods can model ‘real’ materials with dopants, grain boundaries, and interfaces with acceptable accuracy, the high demand for computational resources no longer meets the needs of modern scientific research. On the other hand, Machine Learning (ML) method can accelerate the screening of alloy-based catalytic materials. In this study, an ML model was developed to predict the CO 2 and CO adsorption affinity on single-atom doped binary alloys based on the thermochemical properties of component metals. By using a greedy algorithm, the best combination of features was determined, and the ML model was trained and verified based on a data set containing 78 alloys on which the adsorption energy values of CO 2 and CO were calculated from DFT. Comparison between predicted and DFT calculated adsorption energy values suggests that the extreme gradient boosting (XGBoost) algorithm has excellent generalization performance, and the R -squared ( R 2 ) for CO 2 and CO adsorption energy prediction are 0.96 and 0.91, respectively. The errors of predicted adsorption energy are 0.138 eV and 0.075 eV for CO 2 and CO, respectively. This model can be expected to advance our understanding of structure–property relationships at the fundamental level and be used in large-scale screening of alloy-based catalysts.

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A prediction model for CO 2 /CO adsorption performance on binary alloys based on machine learning

X. Cao, W. Luo and H. Liu, RSC Adv. , 2024,  14 , 12235 DOI: 10.1039/D4RA00710G

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Developing a machine learning model to explore DNA methylation

by Olivia Dimmer, Northwestern University

A Northwestern Medicine study has detailed the development of a machine learning model to predict DNA methylation status in cell-free DNA by its fragmentation patterns, according to findings published in Nature Communications .

DNA methylation, the biological process by which methyl groups are added to a DNA molecule, functions as an "off switch" for certain genes and is commonly dysfunctional in diseases such as cancer.

Cell-free DNA—small amounts of DNA leftover from various cellular processes—can be measured by whole-genome bisulfite sequencing, the current gold standard, but an imperfect process that can damage the DNA being sequenced, limiting scientists' ability to study it.

"Cell-free DNA are these short DNA fragments: When a cell is dying, it will release the DNA to the blood," said Yaping Liu, Ph.D., assistant professor of Biochemistry and Molecular Genetics, who was first and a co-corresponding author of the study. "This cell-free DNA, which is outside the cell, represents the cell death signatures."

Unlike normal DNA, cell-free DNA breaks apart in specific patterns and is highly correlated with the epigenetic status, which led Liu to wonder if he could use cell-free DNA fragmentation patterns to predict the levels of DNA methylation, he said.

In the study, Liu and his collaborators trained an unsupervised machine learning model to analyze small sections of DNA, called CpG sites, using characteristics from the circulating cell-free DNA fragments.

The investigators then used the model to analyze human blood samples from healthy patients and those with different types of cancer and performed separate whole-genome sequencing on the samples to compare the model's accuracy.

The model accurately predicted DNA methylation status mostly at the CpG rich regions on the genome compared to traditional sequencing, according to the study.

"Clinicians already generate a lot of cell-free DNA genomic sequencing data with tests available today," Liu said. "With our model, we can do more with that data and predict DNA methylation and the changes happening in our genes."

The model could also accurately predict which tissues the cell-free DNA came from, thereby pinpointing the origin of abnormal methylation signatures which occur in various cancers, Liu said.

Moving forward, Liu's laboratory will continue to develop computational methods to better understand gene regulation information from cell-free DNA fragments, he said.

"Our goal is to use the epigenetic information hidden in the cell-free DNA to understand the non-coding regions of the human genome," said Liu, who is also a member of the Robert H. Lurie Comprehensive Cancer Center of Northwestern University. "We want to not only detect disease earlier but also get the opportunity to understand what's happening in the genome at that time point."

Journal information: Nature Communications

Provided by Northwestern University

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