Statistical Analysis Link Limited

Statistical Analysis Link Limited A Research Consultancy Firm for Data Collection, Validation and Analysis.

Polynomial regression might seem different from linear regression at first glance, but it’s still considered a linear m...
11/02/2025



Polynomial regression might seem different from linear regression at first glance, but it’s still considered a linear model. Why? It all comes down to how the model parameters are used.

✔️ Linear in Parameters: In polynomial regression, the model remains linear in terms of its coefficients. Even if we include terms like squared or cubic versions of the input variable, the relationship with the parameters stays linear.

✔️ Transformed Features: Instead of treating the data as simple inputs, polynomial regression transforms the original input features (e.g., turning an input into its squared or cubic form). However, the relationship between the coefficients and the target variable remains linear.

✔️ Optimization Stays Linear: The method for estimating the coefficients, such as Ordinary Least Squares, remains the same because the relationship with the parameters does not become non-linear.

❌ Non-Linear Model? If a regression model involves coefficients in non-linear ways, such as multiplying them together or applying complex functions to them, it’s no longer considered linear regression.

When Data is Skewed, Log Transformations Save the Day!Many real-world datasets have highly skewed distributions—think i...
04/02/2025



When Data is Skewed, Log Transformations Save the Day!

Many real-world datasets have highly skewed distributions—think income, population sizes, or transaction amounts. When data spans multiple orders of magnitude, standard statistical models struggle to interpret trends. That’s where log transformations come in!

🔍 Why Use a Log Transformation?
✅ Normalizes Skewed Data – Converts a right-skewed distribution into something closer to normal.
✅ Stabilizes Variance – Reduces heteroscedasticity, making relationships clearer in regression models.
✅ Improves Interpretability – Turns multiplicative effects into additive ones, which simplifies analysis.

🔢 Real-World Example: Income Distribution
I simulated income data following a log-normal distribution (as seen in real earnings data).
• The left plot shows the raw data, which is heavily skewed.
• The right plot shows the log-transformed data, bringing it closer to normal.

To perform a log transformation on skewed data in , use the 'generate' command with the "ln()" function, which calculates the natural logarithm of a variable, effectively transforming it to a log scale; e.g., to transform a variable 'income' into a log-transformed variable called 'log_income', you would use the code:-

gen log_income = ln(income)

Analyze the log-transformed variable in the model

regress outcome_var log_income var2 var3 ....

29/01/2025



Mean imputation is a common method for handling missing values in numerical data. It replaces missing values with the mean of the observed values, ensuring the data set remains complete and easy to use. But is it truly a good option?

Why choose mean imputation:

✔️ Simple and quick to implement, requiring minimal computation.
✔️ Preserves the size of the data set, avoiding the loss of valuable rows.

What to be cautious about:

❌ Replacing missing values with the mean can reduce the natural variability in the data, leading to biased estimates in analyses.
❌ Can distort relationships between variables, particularly in predictive models.
❌ Fails to account for the uncertainty introduced by missing values, which can impact the validity of statistical inferences.

Are There Better Alternatives?

Predictive mean matching is a highly effective alternative to mean imputation. This method identifies observed values closest to the predicted value for a missing entry and randomly selects one of these matches as the replacement. It preserves the natural variability in the data, making it a more robust approach compared to simple mean imputation. Predictive mean matching also works well in handling outliers and maintaining the integrity of relationships between variables. To further enhance reliability and robustness, it is recommended to create multiple imputations. This approach creates several plausible data sets, accounts for the uncertainty in the imputation process, and produces more dependable results for subsequent analyses.

The image below illustrates the impact of mean imputation. The black line represents the original data distribution before imputation, while the red line shows the data distribution after imputation. Notice how mean imputation narrows the distribution, reducing variability and potentially impacting analyses that rely on the data's original structure. For cases where preserving the data's variability is essential, alternative imputation methods should be considered.


mi impute mvn y x1 x2, add(5)

Fitting the model
mi estimate: regress y x1 x2


mi impute pmm x1 x2 x3 x4 x5 x6, add(20) knn(1)

Everyone discusses machine learning, artificial intelligence, deep learning, and generative AI.But we ignore the fundam...
22/01/2025



Everyone discusses machine learning, artificial intelligence, deep learning, and generative AI.

But we ignore the fundamentals of mathematics and statistics.

There was one quote on Twitter that says

“𝘞𝘩𝘦𝘯 𝘺𝘰𝘶’𝘳𝘦 𝘧𝘶𝘯𝘥𝘳𝘢𝘪𝘴𝘪𝘯𝘨, 𝘪𝘵’𝘴 𝘈𝘐. 𝘞𝘩𝘦𝘯 𝘺𝘰𝘶’𝘳𝘦 𝘩𝘪𝘳𝘪𝘯𝘨, 𝘪𝘵’𝘴 𝘔𝘓. 𝘞𝘩𝘦𝘯 𝘺𝘰𝘶’𝘳𝘦 𝘪𝘮𝘱𝘭𝘦𝘮𝘦𝘯𝘵𝘪𝘯𝘨, 𝘪𝘵’𝘴 𝘭𝘰𝘨𝘪𝘴𝘵𝘪𝘤 𝘳𝘦𝘨𝘳𝘦𝘴𝘴𝘪𝘰𝘯.”

Many machine learning algorithms would require a background in statistics and probability. Without it, your model wouldn’t know what to do with data.

You don’t need to be a statistics expert to learn ML, but to understand it makes a big difference.

It helps:

* Asking questions about data
* Cleaning and preprocessing data
* Selecting the right features
* Model evaluation
* Model prediction

When dealing with large datasets, it helps to summarise and simplify complex patterns, so we can actually make sense of them.

Influential Observations are data points that can disproportionately impact the results of a regression model. This plo...
09/01/2025



Influential Observations are data points that can disproportionately impact the results of a regression model. This plot shows the relationship between standardized residuals and leverage, with Cook's distance contours to identify such points.

Why are Influential Observations Important?

Influential data points can distort the model’s fit, leading to incorrect estimates or predictions. Identifying and evaluating these points ensures that the model remains reliable and robust.

How to Interpret the Plot:

🔹 Leverage (x-axis): Measures how far an observation's predictor values are from the mean of the predictor values. Points with high leverage are farther from the center of the predictor space and can have more influence on the regression model.
🔹 Standardized Residuals (y-axis): Shows how far the observed value is from the predicted value, measured in standardized units. Large residuals suggest that the model does not fit the corresponding observation well.
🔹 Cook’s Distance Contours (dashed lines): Cook's distance measures the overall influence of each observation on the fitted regression model. Observations with a Cook's distance greater than 1 are considered potentially highly influential.

✔️ Points within the contour lines (Cook's distance < 0.5): These observations have relatively low influence on the model.
❌ Points near or beyond the contour lines (Cook's distance > 0.5 or > 1): Observations outside or near the dashed lines, such as point 85 in this plot, are considered influential and should be reviewed. They may disproportionately affect the regression model’s coefficients and predictions.

In this plot, most observations are well within the Cook's distance contour lines, indicating low influence on the model.



preserve
gen obs =_n

probit foreign mpg weight
keep if e(sample)

jackknife, saving(b_replic, replace): probit foreign mpg weight

assert e(N_misreps) ==0
mmerge _n using b_replic.dta

describe, fullnames

foreach var in mpg weight {
2. gen dfbeta_`var' = (_b[`var'] -foreign_b_`var')
3. }

gen dfbeta_cons = (_b[_cons] - foreign_b_cons)

label var obs "observation number"
label var dfbeta_mpg "dfbeta for mpg"
label var dfbeta_weight "dfbeta for weight"
label var dfbeta_cons "dfbeta for the constant"

scatter dfbeta_mpg obs, mlabel(obs) title("dfbeta values for variable mpg")

restore

Error handling isn’t just about catching exceptions; it’s about planning for the unexpected.Error management in Softwar...
29/12/2024



Error handling isn’t just about catching exceptions; it’s about planning for the unexpected.

Error management in Software should :

👉 Use Specific Exceptions
• Create custom exceptions for clarity and better debugging.

👉 Centralize Error Handling
• Use middleware or decorators for consistent error handling.

👉 Log Meaningfully
• Log useful details, avoiding sensitive data.

👉 Validate Inputs
• Validate inputs early to catch errors quickly.

👉 Fail Gracefully
• Provide user-friendly error messages without exposing stack traces.

👉 Retry Mechanisms
• Implement retries with backoff for transient errors.

👉 Monitor Errors in Real-Time
• Track and alert errors as they happen.

👉 Use Circuit Breakers
• Stop retries after a set number of failures to prevent overload.

👉 Write Comprehensive Tests
• Cover edge cases and error scenarios in tests.

👉 Implement Graceful Degradation
• Keep core features running when non-critical ones fail.

👉 Log Error Metrics
• Track error trends and prioritize fixes.

Bootstrapping is a powerful statistical method used to estimate the distribution of a statistic by resampling with repl...
14/11/2024



Bootstrapping is a powerful statistical method used to estimate the distribution of a statistic by resampling with replacement from a single sample.

It allows you to make inferences about a population even when you have limited data.

Challenges:
❌ Computational Cost: Can be resource-intensive as it involves multiple resampling iterations.
❌ Bias: Results can be biased if the initial sample is not representative of the population.
❌ Variance: High variance in estimates can occur, especially with small sample sizes.

Advantages:
✔️ Versatility: Bootstrapping can be applied to a wide range of statistics and is especially useful when the theoretical distribution is unknown.
✔️ No Assumptions Needed: It doesn’t require assumptions about the shape of the population distribution.
✔️ Robustness: Provides reliable estimates even with small sample sizes.

In practice, bootstrapping can be implemented in



bootstrap r(p50), reps(1000): summarize var, detail

bootstrap r(ratio), reps(1000) seed(4567): myratio

This means that we will execute bootstrap with our myratio program for 1,000 replications and specify a random-number seed so you can reproduce.

The visualization illustrates how a sample is used to generate resamples (orange) through bootstrapping. Points that appear multiple times in the resamples are highlighted in red. This process helps in estimating the distribution of a statistic by examining the histogram of the resampled statistics.

Statistics isn't just about numbers—it's about making sense of data and drawing meaningful conclusions.These are some k...
03/11/2024



Statistics isn't just about numbers—it's about making sense of data and drawing meaningful conclusions.

These are some key concepts that shape our understanding:

➡️ Correlation: Not everything that moves together is connected! Correlation shows a relationship between variables, but correlation ≠ causation. Always dig deeper before jumping to conclusions.

➡️ P-Value: This is your tool for significance. A small p-value (usually

Evaluating ANOVA results is a critical step in statistical analysis to ensure the reliability of your conclusions. Prop...
13/09/2024



Evaluating ANOVA results is a critical step in statistical analysis to ensure the reliability of your conclusions. Properly checking for homogeneity of variances and normality of residuals helps validate the assumptions underlying ANOVA, leading to more accurate and trustworthy results.

✔️ Improved Validity: By ensuring homogeneity of variances, you minimize the risk of false positives, making your conclusions more robust.
✔️ Accurate Inferences: Checking the normality of residuals allows for more precise estimations and predictions, which are crucial in decision-making.

❌ Misleading Results: Ignoring these checks can lead to biased outcomes, where the data set does not meet the necessary assumptions, compromising the integrity of your analysis.
❌ Erroneous Conclusions: Failing to validate assumptions can result in inaccurate inferences, potentially leading to poor decisions based on flawed data interpretations.

Visualization Explanation:

🔹 Check for Homogeneity of Variances (Residuals vs Fitted): This plot helps visualize if the residuals are spread evenly across all fitted values. In this plot, the residuals are not randomly scattered. There seems to be a pattern, especially in the tails, where larger fitted values correspond to higher variance in the residuals. This suggests that the assumption of homogeneity of variances may be violated.
🔹 Check for Normality of Residuals (Normal Q-Q): This plot checks if the residuals follow a normal distribution. In this plot, the residuals deviate from the diagonal line, especially at the tails, indicating they are not normally distributed, which could compromise the validity of the ANOVA results.

10/09/2024


Quantile regression is a valuable tool for analyzing the relationship between variables, especially when data is not evenly distributed or has outliers.

Unlike traditional linear regression, which focuses only on the mean, quantile regression allows us to predict different points across the distribution of the target variable.

Challenges:
❌ Compared to linear regression, quantile regression requires more computational power and can be harder to interpret for non-experts.
❌ Larger sample sizes might be needed to achieve stable and reliable quantile estimates, especially for extreme percentiles.
❌ The model's results might be less intuitive if you are accustomed to traditional regression techniques, which could limit ease of communication.

Advantages:
✔️ Quantile regression helps to explore trends at various quantiles, offering a more detailed picture of your data.
✔️ This method is highly effective for non-normal data, particularly when there are outliers or heavy tails.
✔️ It is ideal for situations where extreme values or various percentiles are as important as the central trend.

How to handle quantile regression in practice:
🔹 Stata: qreg var1 var2 var3, quantile(.25)

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