Standard Linear Regression
Executive Summary

Executive Summary

Model fit and top drivers of Weekly Sales across 200 observations

Observations
200
R Squared
0.99
Adjusted R Squared
0.99
RMSE
9.973
Significant Drivers
4
Top Driver
TV Spend
Linear regression on 200 observations explains 99% of the variation in Weekly Sales (adjusted R² = 0.99) — a strong fit. TV Spend is the strongest driver, followed by Radio Spend. Typical prediction error is 9.973 in the same units as Weekly Sales. 4 driver term(s) are statistically significant at p < 0.05.
Interpretation

Linear regression on 200 observations explains 99% of Weekly Sales variation (R² = 0.99, adjusted 0.99) with typical error of 9.973. TV Spend is the dominant driver, followed by Radio Spend. Four of five driver terms achieve statistical significance at p < 0.05. This strong fit indicates marketing spend channels—particularly TV and Radio—alongside unit price and region, are reliable predictors of weekly sales outcomes in this dataset.

Overview

Analysis Overview

Ordinary least squares regression of Weekly Sales on 5 driver column(s) across 200 observations.

N Observations200
R Squared0.99
Adj R Squared0.99
Rmse9.973
N Drivers5
Top DriverTV Spend
Interpretation

This analysis applies ordinary least squares regression to explain Weekly Sales using 5 drivers: TV Spend, Radio Spend, Social Spend, Unit Price, and Region across 200 observations. The model achieves an R² of 0.99 (adjusted 0.99), meaning 99% of variation in Weekly Sales is captured by these drivers. Each coefficient quantifies that driver's independent effect on Weekly Sales, holding others constant. Typical prediction error (RMSE) is 9.973 in Weekly Sales units. The exceptionally high explanatory power indicates the drivers account for nearly all systematic movement in weekly sales, with minimal unexplained residual variation.

Data Preparation

Data Quality

Missing value handling, driver typing, and exclusions.

Initial Rows200
Final Rows200
Rows Removed0
Interpretation

All 200 rows retained—no observations were excluded due to missing Weekly Sales values. Numeric drivers underwent median imputation where values were missing; categorical variables (Region) were encoded as dummy variables with rare levels grouped as 'Other'. All resulting driver columns proved usable for modelling. Weekly Sales ranged from 34.34 to 462.61 with a median of 244.515, establishing the observational domain within which coefficients apply. No drivers were excluded as constant or identifier-like, ensuring all five specified drivers entered the model.

Visualization

Distribution of the Outcome

How Weekly Sales is distributed across the dataset.

Interpretation

Weekly Sales spans [0,50] to (450,500], with the modal range (300,350] containing 33 observations (16.5% of the total). The distribution is relatively even across the middle ranges: (150,200] holds 32 observations, (200,250] holds 30, and (250,300] holds 31. Only 2 observations fall in the lowest range [0,50], and just 1 in the highest (450,500]. This spread across 200 observations anchors the regression coefficients: they describe expected sales movement within this observed range, not beyond it.

Data Table

Coefficient Estimates

Effect of each driver on Weekly Sales with standard errors, 95% confidence intervals, and significance.

TermEstimateStd ErrorT ValueP ValueCI LowCI HighSignificance
Intercept50.143.28815.256.43e-3543.6556.62***
TV Spend3.1620.0243130.32.46e-1893.1143.21***
Radio Spend1.790.052234.317.63e-841.6871.892***
Social Spend0.89630.031728.252.62e-700.83370.9589***
Unit Price-2.3810.1193-19.961e-48-2.617-2.146***
Region: North-0.55012.145-0.2560.798-4.7813.681
Region: South2.0382.150.9480.344-2.2036.28
Region: West3.0892.121.4570.147-1.0947.271
Interpretation

TV Spend shows the largest effect: a one-unit increase is associated with 3.1621 more in Weekly Sales (95% CI: 3.1143–3.21, p = 2.46e-189). Radio Spend follows with 1.7895 per unit (95% CI: 1.6867–1.8924, p = 7.63e-84), and Social Spend adds 0.8963 per unit (95% CI: 0.8337–0.9589, p = 2.62e-70). Unit Price shows a negative association: −2.3815 per unit increase (95% CI: −2.6169 to −2.1462, p = 1e-48). Region terms (North, South, West) are not statistically significant at p < 0.05, with confidence intervals crossing zero. All three marketing channels show strong, positive, reliable effects on sales.

Visualization

Driver Ranking

Relative statistical influence of each driver, scaled 0-100.

Interpretation

TV Spend dominates with an importance score of 100, reflecting its t-statistic of 130.348 and p-value of 2.46e-189. Radio Spend ranks second at 26.3 (t = 34.313), followed by Social Spend at 21.7 (t = 28.25) and Unit Price at 15.3 (t = −19.963). Region achieves only 1.1 importance with p = 0.147, indicating negligible statistical influence. This ranking shows TV is roughly 3.8 times more influential than Radio and 4.6 times more influential than Social in predicting weekly sales, while geographic location contributes minimally after accounting for spending levels.

Visualization

Top Driver Effect

Observed relationship between the strongest numeric driver and the outcome.

Interpretation

Weekly Sales rises with TV Spend in a strong positive relationship (r = 0.93) across 200 observations. TV Spend ranges from 0.2 to 100. The scatter shows consistent positive trend: lower TV spend values (0.2–10) cluster in lower sales ranges, while higher values (50+) associate with higher sales. The regression coefficient of 3.1621 quantifies this slope. This tight correlation (r = 0.93) and high statistical significance (p = 2.46e-189) establish TV as the primary marketing channel associated with weekly sales variation in this dataset.

Visualization

Predicted vs Actual

Model predictions against observed Weekly Sales.

Interpretation

Model predictions track actual Weekly Sales closely, with points clustering near the diagonal. Predicted values range from 55.545 to 453.8918, matching the observed outcome range. With R² = 0.99, the model captures nearly all systematic variation; residual scatter (typical error 9.973) reflects only minor unmodelled factors. A few predictions show larger mismatches—for instance, predicted 90.2772 versus actual 68.01, and predicted 446.6269 versus actual 427.51—but these are exceptions. The tight fit confirms that TV, Radio, Social spend, Unit Price, and Region together explain weekly sales reliably.

Visualization

Residual Diagnostics

Prediction errors across the fitted range — a check of the linear model's assumptions.

Interpretation

Residuals center near 0 with a spread of 9.998, consistent with the reported RMSE of 9.973. Approximately 50.5% of predictions undershoot actual values, indicating unbiased centering. Residual spread remains stable across the fitted range (from ~55 to ~454), with no evidence of heteroscedasticity—error variance does not grow or shrink with prediction magnitude. This stability supports the linear model's assumptions. A few outliers exist (e.g., residual −22.2672 at fitted 90.2772), but they are rare and do not systematically undermine the model's validity.

Methodology

Methodology

Statistical methodology and diagnostics for Linear Regression — Outcome Drivers

Statistical Method

Linear Regression — Outcome Drivers

Standard-library analysis: explains any numeric outcome from the driver columns you choose, using ordinary least squares regression. Reports coefficient estimates with 95% confidence intervals and significance, ranks drivers by standardized influence, shows the strongest driver's effect, and checks model fit with predicted-vs-actual and residual diagnostics. Works on any dataset: map one numeric outcome and 1-8 drivers (numeric or categorical).

Data
N = 200 observations
Assumptions
  • The outcome is numeric and approximately linear in the drivers
  • Observations are independent
  • Residuals are roughly constant-variance (checked in diagnostics)
Limitations
  • Correlational, not causal — coefficients describe association
  • Non-linear relationships are only partially captured
  • High-cardinality text drivers are grouped or excluded
Software & Citation
MCP Analytics · mcpanalytics.ai
Code Appendix

Analysis Code

Complete R source code for this analysis

Linear Regression — Outcome Drivers

Fits an ordinary least squares regression to explain a numeric outcome from one or more driver columns chosen by the user. Reports coefficient estimates with confidence intervals and significance, ranks drivers by standardized effect, and surfaces model fit and residual diagnostics.

Why This Method?

Linear regression is the workhorse of explanatory analytics: coefficients are directly interpretable ("one unit of X changes Y by b"), significance tests separate signal from noise, and diagnostics reveal when the linear story is incomplete. It is the right first model for almost any "what drives this number?" question.

What This Analysis Covers

  • Outcome distribution
  • Coefficient table with 95% confidence intervals and p-values
  • Driver ranking by standardized effect size
  • Effect of the strongest numeric driver
  • Predicted vs actual and residual diagnostics

Standard Library

This is a platform standard-library module (LAT-1441): it runs on ANY dataset via the semantic mapping {outcome, driver_1..driver_N}. All narrative text is derived from the user's own column names and computed values — nothing dataset-specific is hardcoded.

suppressPackageStartupMessages(library(htmltools))
suppressPackageStartupMessages(library(jsonlite))
suppressPackageStartupMessages(library(plotly))
suppressPackageStartupMessages(library(DT))
suppressPackageStartupMessages(library(htmlwidgets))
suppressPackageStartupMessages(library(arrow))
suppressPackageStartupMessages(library(knitr))
suppressPackageStartupMessages(library(rmarkdown))
suppressPackageStartupMessages(library(dplyr))
suppressPackageStartupMessages(library(tidyr))
suppressPackageStartupMessages(library(ggplot2))
suppressPackageStartupMessages(library(stringr))
suppressPackageStartupMessages(library(lubridate))
suppressPackageStartupMessages(library(broom))
suppressPackageStartupMessages(library(Matrix))
suppressPackageStartupMessages(library(cluster))
suppressPackageStartupMessages(library(data.table))

Step 1: Row accounting + semantic column discovery

initial_rows <- nrow(df)
  if (!"outcome" %in% names(df)) {
    stop("column_mapping must map an &#x27;outcome' column (the numeric value to explain)")
  }
  driver_cols <- grep("^driver_[0-9]+$", names(df), value = TRUE)
  driver_cols <- driver_cols[order(as.integer(sub("^driver_", "", driver_cols)))]
  if (length(driver_cols) == 0) {
    stop("column_mapping must map at least one driver column(driver_1)")
  }

  outcome_name <- humanize_semantic("outcome", col_map)
  driver_names <- setNames(humanize_semantic(driver_cols, col_map), driver_cols)

Step 2: Coerce outcome to numeric; drop rows with missing outcome

df$outcome <- suppressWarnings(as.numeric(df$outcome))
  df <- df[!is.na(df$outcome), , drop = FALSE]
  if (nrow(df) < 10) {
    stop(sprintf(
      "Only %d rows have a usable numeric value in the outcome column(&#x27;%s'). At least 10 are required.",
      nrow(df), outcome_name))
  }

Step 3: Type each driver — numeric if >=95% of values convert, else factor

dropped_drivers <- character(0)
  for (dc in driver_cols) {
    v <- df[[dc]]
    if (!is.numeric(v)) {
      conv <- suppressWarnings(as.numeric(as.character(v)))
      n_orig <- sum(!is.na(v) & as.character(v) != "")
      if (n_orig > 0 && sum(!is.na(conv)) >= 0.95 * n_orig) {
        df[[dc]] <- conv
      }
    }
    v <- df[[dc]]
    if (is.numeric(v)) {

Numeric: impute NA with median

med <- median(v, na.rm = TRUE)
      if (is.na(med)) { dropped_drivers <- c(dropped_drivers, dc); next }
      v[is.na(v)] <- med
      df[[dc]] <- v
    } else {

Categorical: blank/NA -> "Missing"; lump beyond 12 levels into "Other"

v <- as.character(v)
      v[is.na(v) | trimws(v) == ""] <- "Missing"
      tab <- sort(table(v), decreasing = TRUE)
      if (length(tab) > 12) {
        keep <- names(tab)[1:12]
        v[!(v %in% keep)] <- "Other"
      }
      if (length(unique(v)) > nrow(df) / 2) {

Near-unique text column (an ID, not a driver) — exclude

dropped_drivers <- c(dropped_drivers, dc)
        next
      }
      df[[dc]] <- factor(v)
    }
  }

Step 4: Drop zero-variance drivers

for (dc in setdiff(driver_cols, dropped_drivers)) {
    v <- df[[dc]]
    zero_var <- if (is.numeric(v)) {
      isTRUE(var(v, na.rm = TRUE) == 0) || is.na(var(v, na.rm = TRUE))
    } else {
      length(unique(v)) <= 1
    }
    if (zero_var) dropped_drivers <- c(dropped_drivers, dc)
  }
  model_drivers <- setdiff(driver_cols, dropped_drivers)
  if (length(model_drivers) == 0) {
    stop("No usable driver columns remained after cleaning(all were constant, empty, or identifier-like).")
  }

  df_clean <- df[, c(model_drivers, "outcome"), drop = FALSE]
  final_rows <- nrow(df_clean)
  rows_removed <- initial_rows - final_rows

Step 5: Guard — need clearly more rows than model terms

n_terms_est <- sum(sapply(model_drivers, function(dc) {
    if (is.factor(df_clean[[dc]])) nlevels(df_clean[[dc]]) - 1 else 1
  }))
  while (n_terms_est >= final_rows - 5 && length(model_drivers) > 1) {
    drop_dc <- model_drivers[length(model_drivers)]
    dropped_drivers <- c(dropped_drivers, drop_dc)
    model_drivers <- model_drivers[-length(model_drivers)]
    df_clean <- df_clean[, c(model_drivers, "outcome"), drop = FALSE]
    n_terms_est <- sum(sapply(model_drivers, function(dc) {
      if (is.factor(df_clean[[dc]])) nlevels(df_clean[[dc]]) - 1 else 1
    }))
  }

Step 6: Fit OLS

model <- lm(outcome ~ ., data = df_clean)
  glance_df <- as.data.frame(broom::glance(model))

Step 7: Coefficient table with 95% CIs and significance stars

tidy_df <- broom::tidy(model, conf.int = TRUE, conf.level = 0.95)
  coefficients_df <- data.frame(
    term         = humanize_term(tidy_df$term, model_drivers, col_map),
    estimate     = round(tidy_df$estimate, 4),
    std_error    = round(tidy_df$std.error, 4),
    t_value      = round(tidy_df$statistic, 3),
    p_value      = signif(tidy_df$p.value, 3),
    ci_low       = round(tidy_df$conf.low, 4),
    ci_high      = round(tidy_df$conf.high, 4),
    significance = ifelse(is.na(tidy_df$p.value), "redundant",
                   ifelse(tidy_df$p.value < 0.001, "***",
                   ifelse(tidy_df$p.value < 0.01,  "**",
                   ifelse(tidy_df$p.value < 0.05,  "*",
                   ifelse(tidy_df$p.value < 0.1,   ".", ""))))),
    stringsAsFactors = FALSE
  )

Step 8: Driver importance — max |t| across a driver's terms, 0-100 scale

Direction from the sign of the strongest term (numeric drivers only).

term_raw <- tidy_df$term[tidy_df$term != "(Intercept)"]
  term_t   <- tidy_df$statistic[tidy_df$term != "(Intercept)"]
  term_est <- tidy_df$estimate[tidy_df$term != "(Intercept)"]
  term_p   <- tidy_df$p.value[tidy_df$term != "(Intercept)"]
  imp_rows <- lapply(model_drivers, function(dc) {
    idx <- which(startsWith(term_raw, dc))
    # Aliased (perfectly collinear) drivers have NA statistics — no
    # estimable independent effect; exclude from the ranking instead of
    # crashing on which.max(all-NA) -> integer(0).
    idx <- idx[!is.na(term_t[idx])]
    if (length(idx) == 0) return(NULL)
    best <- idx[which.max(abs(term_t[idx]))]
    data.frame(
      semantic  = dc,
      driver    = driver_names[[dc]],
      abs_t     = abs(term_t[best]),
      direction = if (is.numeric(df_clean[[dc]])) {
        if (term_est[best] > 0) "positive" else "negative"
      } else "categorical",
      p_value   = signif(term_p[best], 3),
      stringsAsFactors = FALSE
    )
  })
  importance_df <- do.call(rbind, Filter(Negate(is.null), imp_rows))
  if (!is.null(importance_df) && nrow(importance_df) > 0) {
    importance_df <- importance_df[order(-importance_df$abs_t), , drop = FALSE]
    max_t <- max(importance_df$abs_t, na.rm = TRUE)
    importance_df$importance_score <- if (is.finite(max_t) && max_t > 0) {
      round(100 * importance_df$abs_t / max_t, 1)
    } else 0
    rownames(importance_df) <- NULL
  } else {
    importance_df <- data.frame(semantic = character(0), driver = character(0),
                                abs_t = numeric(0), direction = character(0),
                                p_value = numeric(0), importance_score = numeric(0),
                                stringsAsFactors = FALSE)
  }
  top_driver_name <- if (nrow(importance_df) > 0) importance_df$driver[1] else "Unknown"
  top_driver_sem  <- if (nrow(importance_df) > 0) importance_df$semantic[1] else NA_character_

Step 9: Outcome distribution — 7 pretty bins

brks <- pretty(range(df_clean$outcome, na.rm = TRUE), n = 7)
  bins <- cut(df_clean$outcome, breaks = brks, include.lowest = TRUE, dig.lab = 10)
  dist_tbl <- table(bins)
  outcome_distribution_df <- data.frame(
    outcome_range = as.character(names(dist_tbl)),
    count         = as.integer(dist_tbl),
    stringsAsFactors = FALSE
  )

Step 10: Top numeric driver effect scatter — <=1000 rows

numeric_drivers_ranked <- importance_df$semantic[
    sapply(importance_df$semantic, function(s) is.numeric(df_clean[[s]]))]
  top_numeric_sem <- if (length(numeric_drivers_ranked) > 0) numeric_drivers_ranked[1] else NA_character_
  top_numeric_driver <- if (!is.na(top_numeric_sem)) driver_names[[top_numeric_sem]] else NA_character_
  if (!is.na(top_numeric_sem)) {
    set.seed(42)
    idx <- if (nrow(df_clean) > 1000) sample(nrow(df_clean), 1000) else seq_len(nrow(df_clean))
    top_driver_effect_df <- data.frame(
      driver_value = df_clean[[top_numeric_sem]][idx],
      outcome      = df_clean$outcome[idx],
      stringsAsFactors = FALSE
    )
    top_driver_effect_df <- top_driver_effect_df[order(top_driver_effect_df$driver_value), ]
    rownames(top_driver_effect_df) <- NULL
  } else {
    top_driver_effect_df <- data.frame(driver_value = numeric(0), outcome = numeric(0),
                                       stringsAsFactors = FALSE)
  }

Step 11: Predicted vs actual and residuals — <=1000 samples

fitted_vals <- fitted(model)
  resid_vals  <- residuals(model)
  set.seed(42)
  ridx <- if (length(fitted_vals) > 1000) sample(length(fitted_vals), 1000) else seq_along(fitted_vals)
  predicted_actual_df <- data.frame(
    predicted = round(fitted_vals[ridx], 4),
    actual    = round(df_clean$outcome[ridx], 4),
    stringsAsFactors = FALSE
  )
  rownames(predicted_actual_df) <- NULL
  residuals_df <- data.frame(
    fitted   = round(fitted_vals[ridx], 4),
    residual = round(resid_vals[ridx], 4),
    stringsAsFactors = FALSE
  )
  rownames(residuals_df) <- NULL

Step 12: KPI metrics (user-facing keys)

r2  <- glance_df$r.squared[1]
  ar2 <- glance_df$adj.r.squared[1]
  rmse <- sqrt(mean(resid_vals^2, na.rm = TRUE))
  n_sig <- sum(coefficients_df$p_value < 0.05 & coefficients_df$term != "Intercept", na.rm = TRUE)
  metrics <- list(
    `Observations`          = final_rows,
    `R Squared`             = round(r2, 3),
    `Adjusted R Squared`    = round(ar2, 3),
    `RMSE`                  = round(rmse, 3),
    `Significant Drivers`   = n_sig,
    `Top Driver`            = top_driver_name
  )

Step 13: json_output machine channel

json_output <- list(
    answer = paste0(
      "Linear regression of ", outcome_name, " on ",
      length(model_drivers), " driver(s) across ",
      format(final_rows, big.mark = ","), " rows: R² = ", round(r2, 3),
      " (adjusted ", round(ar2, 3), "), RMSE = ", round(rmse, 3), ". ",
      top_driver_name, " is the strongest driver(", n_sig,
      " of ", nrow(coefficients_df) - 1, " terms significant at p<0.05)."
    ),
    cards = lapply(
      c("tldr", "overview", "preprocessing", "outcome_distribution",
        "coefficients_table", "driver_importance", "top_driver_effect",
        "predicted_vs_actual", "residual_diagnostics"),
      function(cid) list(id = cid, metrics = metrics)
    )
  )

  list(
    initial_rows            = initial_rows,
    final_rows              = final_rows,
    rows_removed            = rows_removed,
    outcome_name            = outcome_name,
    driver_names            = driver_names,
    model_drivers           = model_drivers,
    dropped_drivers         = dropped_drivers,
    df_clean                = df_clean,
    model                   = model,
    glance_df               = glance_df,
    coefficients_df         = coefficients_df,
    importance_df           = importance_df,
    top_driver_name         = top_driver_name,
    top_driver_sem          = top_driver_sem,
    outcome_distribution_df = outcome_distribution_df,
    top_driver_effect_df    = top_driver_effect_df,
    top_numeric_driver      = top_numeric_driver,
    predicted_actual_df     = predicted_actual_df,
    residuals_df            = residuals_df,
    rmse                    = rmse,
    metrics                 = metrics,
    json_output             = json_output
  )
}

Compute shared resources

shared <- compute_shared(df, params, col_map)

Finalize (do not modify)

This is the exact R code executed to produce the analysis above. Run it with the same dataset to reproduce these results.

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