P Value is a probability score that is used in statistical tests to establish the statistical significance of an observed effect. Though pvalues are commonly used, the definition and meaning is often not very clear even to experienced Statisticians and Data Scientists. In this post I will attempt to explain the intuition behind pvalue as clear as possible.
Content

 Introduction
 When and how is pvalue used?
 What pvalue really is
 How is pvalue used to establish statistical significance
 Practical Guidelines to set the cutoff of Statistical Significance (alpha level)
 What P Value is Not
 Python Example: How to find pvalue for linear regression
 Example: How to find pvalue for linear regression
 Conclusion
1. Introduction
In Data Science interviews, one of the frequently asked questions is ‘What is PValue?”.
According to American Statistical Association,
“a pvalue is the probability under a specified statistical model that a statistical summary of the data (e.g., the sample mean difference between two compared groups) would be equal to or more extreme than its observed value.”
That’s hard to grasp, yes?
Alright, lets understand what really is p value in small meaningful pieces to make it very clear.
2. When and how is pvalue used?
To understand pvalue, you need to understand some background and context behind it. So, let’s start with the basics.
When and how is pvalue used?
pvalues are often reported whenever you perform a statistical significance test (like ttest, chisquare test etc). These tests typically return a computed test statistic and the associated pvalue. This reported value is used to establish the statistical significance of the relationships being tested.
So, whenever you see a pvalue, there is an associated statistical test.
That means, there is a Hypothesis testing being conducted with a defined Null Hypothesis (H0) and a corresponding Alternate hypothesis (HA).
The pvalue reported is used to make a decision on whether the null hypothesis being tested can be rejected or not.
Let’s understand a little bit more about the null and alternate hypothesis.
Now, how to frame a Null hypothesis in general?
While the null hypothesis itself changes with every statistical test, there is a general principle to frame it:
The null hypothesis assumes there is ‘no effect’ or ‘relationship’ by default.
For example: if you are testing if a drug treatment is effective or not, then the null hypothesis will assume there is not difference in outcome between the treated and untreated groups. Likewise, if you are testing if one variable influences another (say, car weight influences the mileage), then null hypothesis will postulate there is no relationship between the two.
It simply implies the absence of an effect.
3. Examples of Statistical Tests reporting out pvalue
Here are some examples of Null hypothesis (H0) for popular statistical tests:
 Welch Two Sample tTest: The true difference in means of two samples is equal to 0
 Linear Regression: The beta coefficient(slope) of the X variable is zero
 Chi Square test: There is no difference between expected frequencies and observed frequencies.
Get the feel?
But how would the alternate hypothesis would look like?
The alternate hypothesis (HA) is always framed to negate the null hypothesis. The corresponding HA for above tests are as follows:
 Welch Two Sample tTest: The true difference in means of two samples is NOT equal to 0
 Linear Regression: The beta coefficient(slope) of the X variable is NOT zero
 Chi Square test: The difference between expected frequencies and observed frequencies is NOT zero.
4. What pvalue really is
Now, back to the discussion on pvalue.
Along with every statistical test, you will get a corresponding pvalue in the results output.
What is this meant for?
It is used to determine if the data is statistically incompatible with the null hypothesis.
Not clear eh?
Let me put it in another way.
The P Value basically helps to answer the question: ‘Does the data really represent the observed effect?’.
This leads us to a more mathematical definition of PValue.
The P Value is the probability of seeing the effect(E) when the null hypothesis is true.
If you think about it, we want this probability to be very low.
Having said that, it is important to remember that pvalue refers to not only what we observed but also observations more extreme than what was observed. That is why the formal definition of pvalue contain the statement ‘would be equal to or more extreme than its observed value.’
5. How is pvalue used to establish statistical significance
Now that you know, p value measures the probability of seeing the effect when the null hypothesis is true.
A sufficiently low value is required to reject the null hypothesis.
Notice how I have used the term ‘Reject the Null Hypothesis’ instead of stating the ‘Alternate Hypothesis is True’.
That’s because, we have tested the effect against the null hypothesis only.
So, when the pvalue is low enough, we reject the null hypothesis and conclude the observed effect holds.
But how low is ‘low enough’ for rejecting the null hypothesis?
This level of ‘low enough’ cutoff is called the alpha level, and you need to decide it before conducting a statistical test.
But how low is ‘low enough’?
6. Practical Guidelines to set the cutoff of Statistical Significance (alpha level)
Let’s first understand what is Alpha level.
It is the cutoff probability for pvalue to establish statistical significance for a given hypothesis test.
For an observed effect to be considered as statistically significant, the pvalue of the test should be lower than the predecided alpha value.
Typically for most statistical tests(but not always), alpha is set as 0.05.
In which case, it has to be less than 0.05 to be considered as statistically significant.
What happens if it is say, 0.051?
It is still considered as not significant. We do NOT call it as a weak statistical significant.
It is either black or white. There is no gray with respect to statistical significance.
Now, how to set the alpha level?
Well, the usual practice is to set it to 0.05.
But when the occurrence of the event is rare, you may want to set a very low alpha. The rarer it is, the lower the alpha.
For example in the CERN’s Hadron collider experiment to detect HiggsBoson particles(which was very rare), the alpha level was set so low to 5 Sigma levels, which means a p value of less than 3 * 10^7 is required reject the null hypothesis.
Whereas for a more likely event, it can go up to 0.1.
Secondly, more the samples (number of observations) you have the lower should be the alpha level. Because, even a small effect can be made to produce a lower pvalue just by increasing the number of observations.
The opposite is also true, that is, a large effect can be made to produce high p value by reducing the sample size.
In case you don’t know how likely the event can occur, its a common practice to set it as 0.05. But, as a thumb rule, never set the alpha greater than 0.1.
Having said that the alpha=0.05 is mostly an arbitrary choice. Then why do most people still use p=0.05?
That’s because thats what is taught in college courses and being traditionally used by the scientific community and publishers.
7. What P Value is Not
Given the uncertainty around the meaning of pvalue, it is very common to misinterpret and use it incorrectly.
Some of the common misconceptions are as follows:
 PValue is the probability of making a mistake. Wrong!
 PValue measures the importance of a variable. Wrong!
 PValue measures the strength of an effect. Wrong!
A smaller pvalue does not signify the variable is more important or even a stronger effect.
Why?
Because, like I mentioned earlier, any effect no matter how small can be made to produce smaller pvalue only by increasing the number of observations (sample size).
Likewise, a larger value does not imply a variable is not important.
For a sound communication, it is necessary to report not just the pvalue but also the sample size along with it. This is especially necessary if the experiments involve different sample sizes.
Secondly, making inferences and business decisions should not be based only on the pvalue being lower than the alpha level.
Analysts should understand the business sense, understand the larger picture and bring out the reasoning before making an inference and not just rely on the pvalue to make the inference for you.
Does this mean the pvalue is not useful anymore?
Not really. It is a useful tool because it provides an objective standard for everyone to assess. Its just that you need to use it the right way.
8. Example: How to find pvalue for linear regression
Linear regression is a traditional statistical modeling algorithm that is used to predict a continuous variable (a.k.a dependent variable) using one or more explanatory variables.
Let’s see an example of extracting the pvalue with linear regression using the mtcars
dataset. In this dataset the specifications of the vehicle and the mileage performance is recorded.
We want to use linear regression to test if one of the specs “the ‘weight’ (wt
) of the vehicle” has a significant relationship (linear) with the ‘mileage’ (mpg
).
This can be conveniently done using python’s statsmodels
library. But first, let’s load the data.
With statsmodels
library
# Load Packages
import pandas as pd
import statsmodels.formula.api as smf
# Import
df = pd.read_csv('https://raw.githubusercontent.com/selva86/datasets/master/mtcars.csv', usecols=['mpg', 'wt'])
df.head()
mpg  wt  

0  4.582576  2.620 
1  4.582576  2.875 
2  4.774935  2.320 
3  4.626013  3.215 
4  4.324350  3.440 
The X(wt
) and Y (mpg
) variables are ready.
Null Hypothesis (H0): The slope of the line of best fit (a.k.a beta coefficient) is zero
Alternate Hypothesis (H1): The beta coefficient is not zero.
To implement the test, use the smf.ols()
function available in the formula.api
of `statsmodels`. You can pass in the formula itself as the first argument and call fit()
to train the linear model.
# Train model
model = smf.ols('mpg ~ wt', data=df).fit()
Once model is trained, call model.summary()
to get a comprehensive view of the statistics.
# Results
print(model.summary())
OLS Regression Results
==============================================================================
Dep. Variable: mpg Rsquared: 0.780
Model: OLS Adj. Rsquared: 0.773
Method: Least Squares Fstatistic: 106.7
Date: Mon, 30 Sep 2019 Prob (Fstatistic): 2.15e11
Time: 14:04:29 LogLikelihood: 7.3950
No. Observations: 32 AIC: 18.79
Df Residuals: 30 BIC: 21.72
Df Model: 1
Covariance Type: nonrobust
==============================================================================
coef std err t P>t [0.025 0.975]

Intercept 6.3553 0.194 32.741 0.000 5.959 6.752
wt 0.5969 0.058 10.328 0.000 0.715 0.479
==============================================================================
Omnibus: 1.783 DurbinWatson: 1.363
Prob(Omnibus): 0.410 JarqueBera (JB): 1.523
Skew: 0.514 Prob(JB): 0.467
Kurtosis: 2.707 Cond. No. 12.7
==============================================================================
Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
The pvalue is located in under the P>t
against wt
row. If you want to extract that value into a variable, use model.pvalues
.
# PValues
print(model.pvalues)
Intercept 4.891527e25
wt 2.146343e11
dtype: float64
Since the pvalue is much lower than the significance level (0.01), we reject the null hypothesis that the slope is zero and take that the data really represents the effect.
Well, that was just one example of computing pvalue.
Whereas pvalue can be associated with numerous statistical tests. If you are interested in finding out more about how it is used, see more examples of statistical tests with pvalues.
9. Conclusion
In this post we covered what exactly is a pvalue and how and how not to use it. We also saw a python example related to computing the pvalue associated with linear regression.
Now with this understanding, let’s conclude what is the difference between Statistical Model from Machine Learning model?
Well, while both statistical as well as machine learning models are associated with making predictions, there can be many differences between these two. But most simply put, any predictive model that has pvalues associated with it are considered as statistical model.
Happy learning!