Logistic Regression: Understanding Statistics

Sancharika

Sancharika

· 5 min read
Logistic Regression

Understanding Statistics

Maximum likelihood

Imagine you have a bag full of colored marbles, but you don't know how many of each color there are. You want to guess how many red marbles are in the bag.

Maximum likelihood is like playing a guessing game with your friend. Here's how:

Pick a guess: You guess a number of red marbles, like 5.

Check the fit: You reach into the bag and pull out some marbles. If you get mostly red marbles, your guess of 5 might be a good one! But if you only get a few red and mostly other colors, your guess might be off.

Make a better guess: Based on what you saw, you try again. Maybe you guess 3 red marbles this time.

Keep guessing: You keep guessing and checking until you find the number that makes it most likely you'd pull out the kind of marbles you just saw (mostly red in this case).

That number you land on, the one that makes the most sense based on what you actually saw, is kind of like the maximum likelihood estimate! It's the guess that best explains the information you have.

This is what people who study science do too, but with more complicated information and fancier math. They use maximum likelihood to figure out things they can't count directly, like how many owls live in a forest or how likely it is to rain tomorrow.

How it works

A peek at how maximum likelihood works with a bit more complexity:

Imagine a Coin Flip Game:

  • You have a coin, but you don't know if it's fair (lands on heads or tails equally) or biased (favors one side).
  • You flip the coin 10 times and get 8 heads.

The Math Part (Not Super Scary!):

  • We can use a formula to calculate the likelihood of getting different results (heads or tails) for each flip. This formula involves a mysterious number called "p," which represents the chance of getting heads on a single flip.
  • If the coin is fair (p = 0.5), the formula tells us how likely it is to get 8 heads and 2 tails in 10 flips.
  • But what if the coin is biased? We can try different values for p (like 0.6 or 0.7) and see how likely it is to get 8 heads with each p value.

Finding the Most Likely "p":

  • The idea is to find the value of p that makes it MOST LIKELY to get the results we actually saw (8 heads). This "most likely p" is the maximum likelihood estimate.
  • We use a special tool called a "likelihood function" that considers all the p values and tells us which one gives the highest likelihood. It's like a scorecard for how well each p explains the data (8 heads).

Remember:

  • This is a simplified example. Real-world maximum likelihood uses more complex math and can involve many variables (not just p).
  • It's a powerful tool for scientists to make informed guesses about things they can't directly measure.

The takeaway: Maximum likelihood helps us find the explanation (like the fairness of the coin) that best fits the evidence we have (like the number of heads flipped).

Interview POV

In an interview setting, you can explain maximum likelihood (MLE) by combining the clarity of the simple explanation with the technical aspects of the explanation. Here's how you can structure your answer:

Start with the basic idea:

  • Briefly introduce MLE as a statistical method to estimate unknown parameters in a model based on observed data.

Use an analogy:

  • Follow up with an analogy like the colored marble bag or the biased coin flip from your previous explanations. This helps the interviewer understand your grasp of the core concept.

Introduce the technical aspects:

  • Mention the concept of a likelihood function. Briefly explain that it represents the probability of observing the data given a specific parameter value.

Highlight the goal:

  • Explain that MLE aims to find the parameter value that maximizes the likelihood function. This value represents the most probable explanation for the observed data.

Optional: Briefly mention limitations:

  • You can add a quick note that MLE might not always lead to a unique solution and there might be multiple possible parameter values.

Here's an example response you can adapt:

"Maximum likelihood estimation, or MLE, is a way to find the most likely explanation for a set of data. Imagine you have a bag of marbles, and you want to know how many red ones there are. You can't count them all, so you pull out a few. MLE is like making an educated guess based on what you see.

In statistics, we use a formula called the likelihood function to calculate the probability of getting the data we observed, given a specific value for an unknown parameter. For example, if we flip a coin and get mostly heads, the likelihood function tells us how likely that is for different probabilities of getting heads (like 50% for a fair coin or 70% for a biased one).

The goal of MLE is to find the value for that unknown parameter (like the coin's bias) that makes it most likely we would have gotten the data we saw (mostly heads). It's like finding the best-fitting explanation for the evidence."

By combining the core concepts with relevant technical terms, you can demonstrate a clear understanding of maximum likelihood estimation in an interview setting.

Models

Null Model

Imagine you're a detective and you need to solve a mystery! You have a crime scene with lots of clues, but you don't know if they're connected or just random stuff.

A null model is like having a special tool that helps you figure out if the clues are actually important. Here's how it works:

The "Nothing Happened" Scenario: The null model pretends like nothing interesting happened at the crime scene. It's like erasing all the clues and saying, "Maybe there wasn't even a crime!"

Creating Randomness: The null model then creates a bunch of fake crime scenes with random stuff scattered around, just like the real scene might have been by chance.

Comparing the Mess: By comparing the real scene with the fake ones, the detective can see if the real clues are truly special or just something you might find anywhere.

  • Lots of Fake Matches: If the real clues appear in many of the fake scenes too, then they might not be that important. It's like finding a bunch of candy wrappers at every scene – not a big deal.
  • Unique Clues Stand Out: But if the real clues are very different from the fake scenes, then they might be real evidence! It's like finding a secret code only at the real crime scene – that's a hot lead!

The Takeaway:

The null model helps us see if things are happening by chance or if there's a real cause behind them. It's like a detective's tool to separate the important clues from the random noise!

How it works

Null models use some probability tricks to compare the real world with a world of "no difference." Here's a peek at the math behind them:

Imagine we're studying friendships in a class:

  • We notice some students seem to be closer friends than others.

The Null Model Math (Not too scary!):

Random Friend Shuffling: The null model shuffles the students' names randomly many times. This creates a bunch of fake classrooms where friendships are purely random chance.

Counting Connections: In each fake classroom, we count how many "friendships" appear by chance. This gives us an idea of how many friendships we might expect just from random seating or coincidence.

Comparing the Real vs Fake: We compare the number of friendships in the real class to the average number of friendships in the fake classrooms.

The Math Tools:

  • We might use statistics like averages or proportions to compare the real class to the random ones.
  • More complex null models might involve fancy probability distributions to create the random shuffles.

What do the results tell us?

  • Lots of Random Friendships: If the real class has about the same number of friendships as the fake ones, then the friendships we saw might be due to chance. Maybe the classroom layout or seating chart encourages some interactions more than others.
  • Significantly More Friendships: But if the real class has many more friendships than expected by chance, then there might be something more going on! Maybe students with similar interests are naturally drawn together.

Remember:

  • The specific math tools depend on the situation being studied.
  • Null models are a helpful first step, but they don't tell the whole story. They just help us see if our observations are surprising or just random chance.

The takeaway: Null models use randomization and probability to create a null hypothesis (no difference) and compare it to the real world. This helps us understand if patterns we see are truly special or just random noise.

Interview POV

In an interview setting, you can explain null models by combining the simple explanation with the underlying statistical concepts. Here's how to structure your answer:

Start with the core concept: Briefly define a null model as a statistical method used to assess the significance of an observed effect.

Use an analogy: Use the detective and crime scene analogy from your previous explanation. This helps the interviewer understand your grasp of the basic idea.

Explain the randomization process: Explain how a null model creates a scenario where the observed effect is absent. This could involve random shuffling, resampling, or other techniques depending on the context.

Highlight the comparison: Explain that the null model compares the observed data (real crime scene) with the data generated under the null hypothesis (fake crime scenes). This helps assess if the observed effect is likely due to chance or a real phenomenon.

Mention statistical tools (Optional): Briefly mention common statistical tools used with null models, like p-values or confidence intervals. You don't need to go into deep mathematical details.

Here's an example response you can adapt:

"A null model is a statistical tool that helps us understand if an observed effect is due to chance or a real underlying cause. Imagine a detective investigating a crime scene with seemingly suspicious clues. They might use a null model like randomly rearranging objects in the room to see if those same 'clues' would appear by pure coincidence.

In statistics, we create a scenario where the effect we're interested in isn't there (the null hypothesis). We then compare the actual data (real crime scene) with data generated under the null model (randomly rearranged rooms). This comparison helps us see if the observed effect is statistically significant or just something we might expect by chance alone.

Statistical tools like p-values or confidence intervals help us quantify the likelihood that the observed effect could be due to random chance. A low p-value or a narrow confidence interval that doesn't include zero suggests the observed effect is unlikely to be random."

By combining the core concept with relevant technical terms, you can demonstrate a clear understanding of null models in an interview setting.

Saturated Model

Imagine you're building a tower with Legos. You have a big box with all sorts of shapes and sizes.

  • A Regular Model: When you build a normal tower, you might use some Legos but not all of them. You pick and choose the ones that fit your design. This is like a regular model in science. You use some measurements or data to explain something, but you might not consider everything.
  • The Saturated Model Tower: Now, imagine building a tower where you use EVERY single Lego in the box! It would be a giant mess, wouldn't it? That's kind of like a saturated model. It uses ALL the information available, fitting every single piece of data perfectly.

Why is a Saturated Model Messy?

  • Too Much Detail: A saturated model might be too specific to the data you have. Imagine if your tower was so specific it wouldn't work with any other Legos!
  • Not Useful for the Future: Since it uses all the data, it might not be good at predicting what happens with new information. It's like a tower so specific to the box it wouldn't work with other Lego sets.

The Takeaway:

Saturated models are like using every single Lego – they fit the data perfectly, but they might be too specific and not very useful for understanding new things.

How it works

In statistics, a saturated model refers to a model that has as many parameters as there are data points. Here's a breakdown:

  • Parameters: These are the unknowns we're trying to estimate in the model, like the height or width of each Lego in our tower.
  • Data Points: These are the actual observations we have, like the number of red Legos or the length of a specific piece.

Why are Saturated Models not ideal?

  • Overfitting: With the same number of parameters as data points, a saturated model perfectly fits the data. But it might be memorizing the specific details of the data rather than capturing the underlying relationships. This is like the Lego tower that only works with that specific box.
  • No Degrees of Freedom: Degrees of freedom represent the ability of a model to generalize beyond the data used to fit it. A saturated model has zero degrees of freedom because all the flexibility is used up fitting the data exactly. It's like the Lego tower being so specific it can't be adapted for anything else.

The Takeaway:

Saturated models, while achieving perfect fit, are not very useful in practice. They lack the ability to generalize and might be overfitting the specific data. We generally prefer models that capture the essential relationships with fewer parameters for better prediction and understanding.

Interview POV

Here's how you can explain a saturated model in an interview setting:

1. Define the Concept:

Start by providing a clear definition: "A saturated model is a statistical model that has the same number of parameters as there are data points."

2. Break down the Key Terms:

Briefly explain what "parameters" and "data points" mean in this context:

  • Parameters: These are the unknown values the model is trying to estimate. Imagine them as the measurements or properties of the Legos in your tower design (height, width etc.).
  • Data Points: These are the actual observations you have. They're like the specific Legos you have in your collection (number of red ones, length of a specific piece).

3. Explain the Implications:

Highlight the key drawbacks of saturated models:

  • Overfitting: Since the model has as many parameters as data points, it can perfectly fit the data. However, this might be because it's memorizing the specific details of the data rather than capturing the underlying relationships. It's like building a Lego tower so specific to the box it wouldn't work with any other Lego sets.
  • No Degrees of Freedom: Degrees of freedom represent the ability of a model to generalize beyond the data used to fit it. A saturated model has zero degrees of freedom because all the flexibility is used up fitting the data exactly. There's no room for the model to adapt to new information, similar to the Lego tower being so specific it can't be built with different Legos.

4. Briefly Mention the Advantage (Optional):

You can very briefly mention one advantage (if relevant to the interview context):

  • Perfect Fit: Saturated models achieve a perfect fit on the data they are trained on. This can be useful in specific situations where the exact replication of the data is the primary goal.

5. Conclude with the Importance of Balance:

Wrap up your explanation by emphasizing the trade-off:

"While saturated models achieve perfect fit, they lack the ability to generalize and might be overfitting the data. In practice, we usually prefer models that capture the essential relationships with fewer parameters for better prediction and understanding."

By combining this structure with clear explanations, you can effectively communicate your understanding of saturated models in an interview setting.

Proposed Model

Imagine you're building a cool fort out of blankets and pillows. You have a plan in your head for how you want it to look (the proposed model).

  • The Plan: This plan might involve using a big blanket for the roof, pillows for walls, and maybe some chairs for support. It's like your best guess for how to build the perfect fort!
  • Building the Fort: Now, when you actually start building, things might not go exactly according to plan. Maybe the blanket isn't big enough, or a pillow falls over. That's okay! You can adjust your plan as you go (revised model).

The Proposed Model is Like a First Guess:

  • In science, a proposed model is like your initial plan for the fort. It's an idea about how something works, based on what we already know. Scientists might use observations, experiments, or even other models to come up with this proposed model.
  • Testing and Refining: Just like you might need to adjust your fort plan, scientists test their proposed models to see if they work in the real world. They might need to change or refine the model based on the results (revised model).

The Takeaway:

A proposed model is a scientist's first guess about how something works. It's like a plan that can be adjusted and improved as they learn more!

How it works

Building the Perfect Sandcastle (Analogy):

Imagine you're at the beach and want to build the most amazing sandcastle ever!

  • The Proposed Model: Your proposed model is your initial plan for the sandcastle. You might sketch it out, envisioning tall towers and intricate details.
  • Building and Measuring (Data Collection): As you start building, you measure the height of your towers and the width of your moat (data collection).
  • Checking How Close You Are (Deviance): Here's where things get interesting. You compare your actual sandcastle measurements to your perfect plan (proposed model). This difference between your ideal and reality is a kind of "deviation."
    • Small Deviation = Great Castle! If the measurements are very close to your plan (small deviation), then you're building a fantastic sandcastle!
    • Large Deviation = Need Adjustments! But if the measurements are way off your plan (large deviation), then something's wrong. Maybe the sand is too wet, or the waves keep knocking things down. You'll need to adjust your plan (revised model) to account for these challenges.

The Math Behind Deviance (Simplified):

Deviance is a statistical measure of how well a proposed model fits the actual data. It uses a formula to calculate the difference between what the model predicts and what's actually observed.

  • Smaller deviance values generally indicate a better fit between the model and the data, similar to your sandcastle measurements being close to your plan.
  • Larger deviance values suggest the model might not be capturing reality well, just like a large difference between your plan and the actual sandcastle.

The Takeaway:

Deviance helps to see how well their proposed models (initial plans) match the real world (the sandcastle you're building). It's like a tool to measure the difference between their best guess and the actual data, allowing them to improve their models over time!

Null Model VS Proposed Model VS Saturated Model

Breakdown of Null Model, Proposed Model, and Saturated Model:

1. Null Model:

  • What it is: The null model represents a scenario where there's no effect of interest. It's like a baseline comparison.
  • Think of it as: A control group in an experiment. Imagine testing a new fertilizer on plants. The null model represents a group of plants that don't receive the fertilizer, so you can see if the fertilizer truly makes a difference.
  • How it's used: It helps assess the significance of an observed effect in the proposed model. If the data in the proposed model is very similar to the null model, then the observed effect might be due to chance, not a real phenomenon.

2. Proposed Model:

  • What it is: The proposed model is the main hypothesis you're testing. It explains how you think things work or what effect you expect to see.
  • Think of it as: A recipe you want to try. You have specific ingredients and steps you believe will lead to a delicious dish (the desired outcome).
  • How it's used: It's the model you build or test to see if it explains the data and supports your hypothesis.

3. Saturated Model:

  • What it is: A saturated model includes every possible variable that could influence the outcome. It fits the data perfectly, but at a cost.
  • Think of it as: Memorizing every detail of a recipe to replicate a dish exactly. While you might get the dish right, it's not very useful for understanding the underlying principles or adapting the recipe for other ingredients.
  • Why it's not ideal: Saturated models can be overly complex and might not generalize well to new data. They might be fitting noise in the data rather than the true relationships.

Here's a table summarizing the key points:

ModelDescriptionAdvantagesDisadvantagesNull ModelNo effect scenarioProvides baseline comparisonDoesn't explain the actual phenomenonProposed ModelMain hypothesis being testedExplains the data and supports your theoryMight be oversimplified or inaccurateSaturated ModelIncludes every possible variablePerfect fit on the data usedOverly complex, prone to overfitting, not generalizable

PS: Ideally, we want the likelihood of the data given our Proposed Model to be larger then the Null Model and close to the Saturated Model.

Interview POV

In an interview setting, explain the relationship between null, proposed, and saturated models by highlighting their key characteristics and how they build upon each other:

Start with Definitions: Briefly define each model:

  • Null Model: "The null model represents a scenario where there's no effect of interest. It's a baseline comparison point."
  • Proposed Model: "The proposed model is the one we're primarily interested in. It captures our hypothesis about the relationship between variables and aims to explain the observed data."
  • Saturated Model: "A saturated model includes every possible variable that could influence the outcome, resulting in a perfect fit to the data used for its creation."

Highlight the Progression: Explain how these models build upon each other:

  • Null Model as the Foundation: "We often start with the null model as a baseline. It helps us assess the significance of the effect observed in the proposed model."
  • Proposed Model as the Hypothesis: "The proposed model is our main hypothesis. It incorporates relevant variables and aims to provide a better explanation for the data compared to the null model (no effect)."
  • Saturated Model for Comparison (Optional): "While not always used, a saturated model can serve as a reference point. It shows the best possible fit achievable with all possible variables. By comparing the deviance (goodness-of-fit) of the proposed model to the null and saturated models, we can understand how well our proposed explanation captures the data compared to no effect and a perfect fit scenario."

Connect to Deviance (Optional): Briefly mention deviance if relevant to the interview context:

  • "Deviance is a statistical measure that tells us how well a model fits the data. By comparing the deviance of the proposed model to the null and saturated models, we gain insights into the effectiveness of our proposed explanation."

Tailor to the Interview Focus: If the interview emphasizes a specific area of study, you can provide a relevant example of how these models are used in that field.

By explaining the relationships between these models and their roles in statistical analysis, you can demonstrate a strong understanding of their applications in an interview setting.

PS: More detailed explaination for Deviance can be found in next Blog !!

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