This document describes the statistical methods used in Phi Statistics Drawer v2. The goal is transparent reporting of the calculations, assumptions, and implementation choices used by each plotting tool.
1. Data Handling
The tools accept either manual table input or imported spreadsheet data. CSV and delimited text files are parsed by detecting tab, semicolon, or comma delimiters. Excel workbooks are read from the first available worksheet. The first row is interpreted as the header row, and duplicate or empty column names are made unique before column mapping.
Numeric entries are parsed after trimming whitespace and accepting a decimal comma or decimal point. Values that cannot be parsed as finite numbers are treated as missing for numeric calculations.
Missing or invalid values are handled by complete-case inclusion for the variables required by the selected tool. In practice, rows with missing required fields are ignored for that calculation and a warning reports the number and type of ignored rows where applicable. The application does not impute missing values and does not silently replace missing values with zero.
2. Kaplan–Meier Survival Analysis
The Kaplan–Meier module implements the standard product-limit estimator, Greenwood variance, and pointwise log–log 95% confidence intervals for right-censored time-to-event data.
2.1 Event Definition
The user selects a time column, a status column, and an event code. A row is treated as an event when the status value matches the configured event code. A non-empty status value that does not match the event code is treated as censored. Rows with missing time, negative time, or missing status are ignored. If no group is supplied, the row is assigned to a default group.
2.2 Kaplan–Meier Estimator
For each group, let \(t_i\) denote follow-up time and \(\delta_i\) the event indicator, where \(\delta_i = 1\) denotes an event and \(\delta_i = 0\) denotes censoring. At each unique event time \(t_j\):
- \(n_j\): number of individuals at risk just before time \(t_j\)
- \(d_j\): number of events at time \(t_j\)
\( S(t_j) = S(t_{j-1}) \cdot \frac{n_j - d_j}{n_j}, \quad S(0) = 1. \)
The curve is drawn as a step function and remains constant between event times. Censored observations are displayed at the survival value active at their censoring time.
2.3 Risk Sets
The risk set at time \(t\) is:
\( n(t) = \#\{ i : \text{follow-up time}_i \ge t \}. \)
This same definition is used for the patients-at-risk table. The table can be shown as a separate table view or embedded below the plot; both use the same risk-set calculation at the displayed time points.
2.4 Greenwood Variance and Log–log Confidence Intervals
Greenwood’s variance estimator is:
\( \mathrm{Var}\bigl(S(t_j)\bigr) = S(t_j)^2 \sum_{k \le j} \frac{d_k}{n_k (n_k - d_k)}. \)
The confidence interval is computed on the log–log scale, similar to the common default in R’s
survival package for survfit. With
\(S_j = S(t_j)\) and \(V_j = \mathrm{Var}(S_j)\):
\( z_j = \log\bigl(-\log(S_j)\bigr), \quad SE(z_j) = \frac{\sqrt{V_j}}{S_j \cdot |\log(S_j)|}. \)
\( z_j \pm 1.96 \cdot SE(z_j). \)
The lower and upper limits are transformed back to the survival scale and constrained to \([0,1]\). Confidence shadows show these pointwise limits as a stepwise band; they are not simultaneous confidence bands.
3. Forest Plot
The forest plot module is a visualization tool for user-supplied point estimates and intervals. It does not estimate odds ratios, risk ratios, hazard ratios, mean differences, or confidence intervals. It also does not perform pooling, weighting, hypothesis testing, or meta-analysis.
3.1 Input Data
For each row, the user supplies a label, a point estimate, a lower interval limit, and an upper interval limit. Rows are ignored if the label is missing, an estimate or interval limit is invalid, or the lower limit is greater than the upper limit.
3.2 Ratio and Difference Scales
The tool supports two effect-type presets:
- Ratio effects: typical for OR, RR, or HR values. The default reference value is 1 and the default x-axis scale is logarithmic.
- Difference effects: typical for mean differences or risk differences. The default reference value is 0 and the default x-axis scale is linear.
Logarithmic axes require strictly positive estimate and interval values. Non-positive values are ignored when a logarithmic axis is selected because they cannot be represented on that scale.
3.3 Plotting Method
The central estimate is plotted as a point symbol. The supplied lower and upper limits are plotted as a horizontal interval line. Optional vertical caps at the ends of the interval are visual markers only and do not alter the interval values.
4. ROC / AUC Analysis
The ROC / AUC tool evaluates one or more numeric marker columns against a binary outcome. The user defines the positive outcome class. Any non-empty outcome value that does not match the positive class is treated as the negative class, so the outcome variable should be binary before analysis.
4.1 ROC Curve
For each marker, thresholds are evaluated across the unique observed marker scores. Depending on the selected direction, a predicted positive result is defined as either:
\( \text{score} \ge c \quad \text{or} \quad \text{score} \le c. \)
At each threshold \(c\), sensitivity and specificity are calculated as:
\( \text{Sensitivity} = \frac{TP}{TP + FN}, \quad \text{Specificity} = \frac{TN}{TN + FP}. \)
The ROC curve plots sensitivity against the false-positive rate \(1 - \text{Specificity}\). The diagonal reference line represents random classification.
4.2 AUC
The AUC is computed as the pairwise probability that a randomly selected positive case has a more positive marker score than a randomly selected negative case according to the selected score direction, with ties contributing 0.5:
\( \mathrm{AUC} = \frac{1}{n_+ n_-} \sum_{i:Y_i=1}\sum_{j:Y_j=0} \psi(s_i, s_j). \)
Here \(\psi = 1\) if the positive case ranks above the negative case after applying the selected score direction, \(\psi = 0.5\) for a tie, and \(\psi = 0\) otherwise. This is equivalent to the Mann–Whitney/Wilcoxon rank interpretation of the AUC.
4.3 Cutoff Selection
The optimal cutoff reported by the tool is selected by maximizing Youden’s index:
\( J = \text{Sensitivity} + \text{Specificity} - 1. \)
If two thresholds have numerically equal Youden index values, the implementation prefers the threshold with higher sensitivity.
4.4 DeLong Statistics and Bootstrap Confidence Intervals
The standard error and p-value for AUC versus 0.5 are calculated using a DeLong-style variance based on placement values for positive and negative cases. The reported p-value uses a two-sided normal approximation:
\( z = \frac{\mathrm{AUC} - 0.5}{SE(\mathrm{AUC})}. \)
When multiple markers are selected, paired DeLong comparisons are reported relative to the first valid marker. These comparisons use only rows with valid scores for both markers being compared.
Bootstrap 95% confidence intervals for AUC are percentile intervals from stratified resampling: positive cases are sampled with replacement from the positive set and negative cases are sampled with replacement from the negative set. The same bootstrap samples are used to draw pointwise ROC confidence bands on a fixed false-positive-rate grid. These ROC bands are pointwise bootstrap bands, not simultaneous confidence bands.
5. Boxplot, Scatterplot, and Fusion Plot
The boxplot tool uses raw numeric values and a grouping variable. For each group, values are sorted and the following descriptive statistics are computed:
- Mean: arithmetic mean
- Median: 50th percentile
- First quartile (Q1): 25th percentile
- Third quartile (Q3): 75th percentile
- Minimum and maximum: smallest and largest observed values
Quartiles are calculated as interpolated empirical quantiles. The box spans Q1 to Q3, with a line at the median. Whiskers extend to the observed minimum and maximum. This is a min-to-max boxplot implementation, not a Tukey 1.5·IQR outlier-rule implementation.
Scatterplot mode displays the raw observations. Fusion mode overlays raw observations and a boxplot. Jitter is a visual separation method for overlapping points and does not change the underlying values. The tool does not perform group comparison tests.
6. Bar Charts
The bar chart tool uses one numeric value column and one group column. Values are aggregated within each group, and each group is represented by one bar. The user can choose the bar height:
- Mean: arithmetic mean of the group values
- Median: 50th percentile of the group values
- Max: largest observed value in the group
- Sum: sum of all valid group values
- Count: number of valid observations in the group
Mean is the default because a bar chart grouped by category is commonly used to display a summary value per category. The summary table always reports \(n\), the selected bar value, mean, median, quartiles, minimum, and maximum.
Optional summary reference lines can display mean, median, and quartiles for the group when the selected bar value is Mean, Median, or Max. The bar chart tool does not calculate standard deviations, standard errors, confidence intervals, or p-values and therefore does not draw inferential error bars.
7. Grouped and Stacked Bar Charts
The grouped and stacked bar chart tools are categorical frequency tools. They use a main group variable and a subgroup variable. No numeric measurement column is used.
For every group \(g\) and subgroup \(s\), the tools compute:
\( n_{g,s} = \#\{ i : \text{group}_i = g,\ \text{subgroup}_i = s \}. \)
If percentage mode is selected, percentages are normalized within each main group:
\( p_{g,s} = 100 \cdot \frac{n_{g,s}}{\sum_s n_{g,s}}. \)
Grouped bar charts display subgroup counts or percentages side by side within each group. Stacked bar charts display subgroup counts or percentages stacked within each group; in percentage mode each stack sums to 100% apart from rounding. These tools are descriptive and do not compute chi-square tests, risk ratios, odds ratios, or confidence intervals.
8. Paired Dot Plot
The paired dot plot visualizes two numeric measurements for the same observational unit, usually a before/after or timepoint 1/timepoint 2 structure. The required fields are ID, group, timepoint 1, and timepoint 2. Rows with missing ID, missing group, or invalid numeric values at either timepoint are ignored.
For each valid row, the change is:
\( \Delta_i = y_{i,2} - y_{i,1}. \)
Individual points show the two measurements for each ID, and optional connecting lines show the direction and magnitude of individual change. Jitter is used only to separate overlapping points visually.
The summary table reports \(n\), mean and median values at both timepoints, and mean and median changes. The optional summary line displays either the group mean or group median at each timepoint. The tool does not perform paired t-tests, Wilcoxon signed-rank tests, confidence intervals, or multiplicity adjustments.
9. Regression Tools
The regression tools fit curves separately for each group using the regression-js
library with high numerical precision. A group requires at least two valid observations and at least
two distinct x-values to produce a fitted curve. The tools report \(n\), \(R^2\), fitted parameters,
and the fitted equation for each group.
The fitted line or curve is sampled for drawing. Display choices such as a logarithmic y-axis change only how the fitted values are plotted; they do not change which regression model is fitted.
9.1 Linear Regression
The linear regression model is:
\( y = a x + b. \)
Slope \(a\) and intercept \(b\) are estimated by ordinary least squares:
\( a = \frac{\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})} {\sum_{i=1}^n (x_i - \bar{x})^2}, \quad b = \bar{y} - a \bar{x}. \)
9.2 Logarithmic Regression
The logarithmic model is:
\( y = a + b \ln(x). \)
Only observations with \(x > 0\) are valid for this model. Rows with \(x \le 0\) are ignored because \(\ln(x)\) is undefined for non-positive values in the real-number domain. The model is fitted by applying linear regression to \((\ln(x_i), y_i)\).
9.3 Exponential Regression
The exponential model is:
\( y = a e^{b x}. \)
Only observations with \(y > 0\) are valid for this model. Rows with \(y \le 0\) are ignored. The
implementation uses the exponential routine from regression-js, which estimates
\(a\) and \(b\) through the library’s closed-form exponential fitting method for positive response
values. This is not a generalized linear model and does not estimate Poisson, logistic, or Cox
regression parameters.
9.4 Coefficient of Determination
For all regression tools, \(R^2\) is computed from observed and fitted values on the original response scale:
\( R^2 = 1 - \frac{\sum_{i=1}^n (y_i - \hat{y}_i)^2} {\sum_{i=1}^n (y_i - \bar{y})^2}. \)
The regression tools are descriptive curve-fitting tools. They do not report standard errors, confidence bands, prediction intervals, p-values, or residual diagnostics.
10. General Remarks
- Displayed values are rounded for readability; internal calculations use unrounded numeric values unless a third-party library returns rounded parameters.
- Annotations, legends, axis formatting, and export settings are visual tools and do not modify the dataset or statistical output.
- The application avoids undocumented smoothing and does not add statistical tests where the selected visualization is descriptive only.
Users remain responsible for verifying that the chosen model, visualization, and data preprocessing are appropriate for their specific data and research question.