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OLS (Ordinary Least Squares) is a fundamental technique used in statistics for linear regression analysis. It aims to find the line (or hyperplane in higher dimensions) that best fits a set of data points by minimizing the sum of the squares of the residuals, which are the differences between observed values and the values predicted by the linear model. The process involves several key steps:

• Model Specification: Defining the dependent and independent variables.
• Parameter Estimation: Calculating the coefficients that minimize the sum of squared residuals.
• Fit Assessment: Evaluating how well the model fits the data, often using metrics like R-squared.

An OLS regression model for factorial analysis is:     
    Assuming we have a dataset containing the fail rates of semiconductor wafers under 40 different test bins (data) as shown below. The dataset includes five columns, each representing one of five wafers, with the fail rates measured for each. The wafers were fabricated using different combinations of 10 possible conditions. Specifically, Wafer1 was fabricated under Conditions 1 and 2; Wafer2 under Conditions 1, 2, 3, 6, and 9; Wafer3 under Conditions 1, 8, 9, and 10; Wafer4 under Conditions 1, 2, 3, 5, and 7; and Wafer5 under Conditions 1, 4, 5, and 8. We want to perform a fail analysis to understand the relationships between these varying fabrication conditions and the observed fail rates across the different bins. This will involve identifying any patterns or correlations that may exist between the conditions and the fail rates, which could help in pinpointing specific conditions that lead to higher fail rates, thereby facilitating improvements in fabrication processes (Script 1):

Any example of input csv data is:

Output from the regression modeling is:    

The output from the modeling is the summary of an OLS (Ordinary Least Squares) regression model, with key components of this output:

• Regression Results Components
  • Dep. Variable: The dependent variable, Wafer1, is the variable being predicted or explained in the model.
  • R-squared: This is 0.000, indicating that the model explains none of the variability of the response data around its mean. In practical terms, the predictors do not explain any variance in the fail rates for Wafer1.
  • Adj. R-squared: Similarly adjusted R-squared, which is also 0.000, indicates no improvement or penalty for the number of predictors used in the model.
  • F-statistic: Shows 'nan' (not a number), which typically indicates an issue such as perfect multicollinearity or other numerical problems in the data that prevent the model from being fit properly.
  • Prob (F-statistic): Also 'nan', indicating that the p-value for the F-statistic can't be computed, which further supports the issue with the model fit.
  • Log-Likelihood: This is a measure of the goodness of fit of the model. The value -163.41 is quite low, indicating poor model fit.
  • AIC (Akaike Information Criterion) and BIC (Bayesian Information Criterion): These are both measures used to compare different models. The lower the values, the better the model fit, relative to other models. Here, they just indicate the model fit in numerical terms without a comparison.
• Coefficients Table
  This table includes the estimated coefficients for each predictor variable in the model, along with their standard errors, t-values, p-values, and confidence intervals.
  • coef: Coefficients represent the change in the dependent variable for a one-unit change in the predictor, all else being equal. Here, each coefficient suggests the impact of a condition on the fail rates.
  • std err: Standard error of the coefficient, indicating the level of uncertainty around the coefficient estimate.
  • t: This is the t-statistic, which is used to determine whether the coefficient is significantly different from zero. A large t-value (in absolute terms) typically indicates that the effect is significant.
  • P>|t|: P-value associated with the t-statistic. A low p-value (< 0.05) suggests that the effect of the predictor is statistically significant.
  • [0.025 0.975]: These are the 95% confidence intervals for the coefficients. If this interval does not include zero, the effect is considered statistically significant at the 5% level.
• Diagnostic Tests
  • Omnibus, Prob(Omnibus), Jarque-Bera (JB), and Prob(JB) are tests for normality of the residuals. Deviation from normality can be a sign of model misspecification.
  • Skew: Measures the symmetry of the data about the mean. Near zero means the data are relatively symmetric.
  • Kurtosis: Measures the tails of the distribution. A value close to 3 (for a normal distribution) is ideal.
  • Durbin-Watson: Tests for autocorrelation in the residuals from a regression. Values from 1.5 to 2.5 generally indicate that there is no autocorrelation.
• Interpretation
  The results suggest significant issues with the model:
  • The R-squared values are zero, indicating no explanatory power.
  • The F-statistic is undefined, which might suggest issues such as multicollinearity.
  • The coefficients seem to be statistically significant, but this might be misleading due to potential issues in model specification or data problems.

To proceed, it's important to revisit the data preparation step, ensure correct encoding of variables, and possibly explore alternative modeling approaches or adjustments to the current model setup.

To understand the impact of each condition on the fail rates from the regression results, we can look at a few key metrics from the outputs. The primary focus should be on the coefficients and their corresponding t-values, and p-values:

• Coefficients
  • Coefficient: This value represents the expected change in the dependent variable (fail rate) for a one-unit change in the predictor variable (presence of a condition), holding all other predictors constant.
  • Std err: This is the standard error of the coefficient, indicating the average distance that the coefficient estimates fall from the actual value.
  • t-value: A higher absolute t-value indicates that the coefficient is significant. It measures the number of standard deviations that the coefficient is away from zero.

• Interpretation Steps
  • Coefficient Significance: Check the p-value associated with each coefficient. A p-value less than 0.05 typically indicates that the coefficient is statistically significant.
  • Magnitude and Direction: The magnitude of the coefficient tells you how much impact the condition has per unit change. The sign tells you the direction of the impact (positive or negative).
  • Compare t-values: Higher absolute t-values suggest a stronger evidence against the null hypothesis (which would state that the condition has no effect). This helps in determining the rank of impacts.

• Example Interpretation:
  • Look at the coefficients and their respective t-values and p-values.
  • Wafer1:
    • ConditionX: Coefficient = 28.0801, t = 24.375, p < 0.000.
    • Condition2: Coefficient = 28.0801, t = 24.375, p < 0.000.

• Highest Impact:
  • Ranking the Impact: Conditions with higher absolute coefficient values and higher t-values are ranked higher in terms of impact.
  • From the outputs above, each wafer is influenced differently by different conditions, but without comparative t-values across conditions for each wafer, it’s difficult to establish a ranking. It appears, however, that all conditions listed in the results are significant.

• Important Notes:
  • Multicollinearity Warning: The warnings about multicollinearity (high condition numbers and issues with the smallest eigenvalue) suggest that the conditions are highly correlated with each other. This can distort the impact estimates and make the coefficients unreliable. We might need to address this by possibly reducing the number of conditions included in the model simultaneously or using techniques like Principal Component Analysis (PCA) or regularization methods in regression.
  • Model Fit Issues: The F-statistics being 'nan' and R-squared values at 0 or negative might indicate issues with the model fit or data issues. It's important to investigate why the model is not fitting well—this might be due to data not meeting the assumptions of linear regression.

Another way, to analyze the impact of different fabrication conditions on the fail rates of the wafers, with OLS regression model is the analysis with correlation analysis and regression models (Script 2). This script does:

• Correlation Analysis: To see if there is any linear relationship between the fail rates and the different wafers. It calculates and visualizes the correlation matrix for the fail rates of different wafers to see how they relate to each other.
• Multiple Linear Regression: To determine the impact of the different conditions on the fail rates. Multiple Linear Regression: The script uses the conditions as independent variables and the mean fail rates of each wafer as the dependent variable to determine the impact of different fabrication conditions. This is done using the statsmodels library.

The linear regression results, including the coefficients and their significance, help identify which conditions have a significant impact on the fail rates. The plots provide a visual comparison between actual and predicted fail rates. The OLS is a method for estimating the unknown parameters in a linear regression model, with the goal of minimizing the sum of the squares of the differences between the observed and predicted values.

In the OLS regression, the script does:

• Preparing the Data:
  • The dataset containing the fail rates for different wafers is prepared.
  • A binary condition matrix is created to represent which fabrication conditions were used for each wafer.

• Correlation Analysis:
  • The correlation matrix is calculated to see the relationships between the fail rates of different wafers.
  • A heatmap is plotted to visualize these correlations.

• Multiple Linear Regression (OLS):
  • The mean fail rates for each wafer are calculated.
  • The binary condition matrix is used as the independent variables (X), and the mean fail rates are used as the dependent variable (y).
  • The statsmodels library is used to perform OLS regression.
  • The model's summary provides details about the coefficients, their significance, and other statistics.

Table 3277 lists the differences between the two scripts.

Table 3277. Differences between the two scripts.

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