update wording around transfer functions

Author: drbenvincent

docs/source/notebooks/graded_intervention_time_series_single_channel_ols.ipynb

@@ -9,25 +9,28 @@
     "**Graded Intervention Time Series** extends classical interrupted time series analysis to handle **graded interventions** - policies or treatments with varying intensity over time, rather than simple on/off changes. Traditional ITS methods model binary interventions (e.g., \"policy enacted\" vs \"no policy\"). This method (technically called Transfer Function Interrupted Time Series or TF-ITS in the literature {cite:p}`box1975intervention`, with extensions to multiple time series by {cite:p}`abraham1980intervention`) handles more realistic scenarios where:\n",
     "\n",
     "1. **Intervention intensity varies continuously** (e.g., advertising spend \\$0 - 100k, communication frequency 0-10 messages/week)\n",
-    "2. **Effects saturate** - diminishing returns as exposure increases (10th message less impactful than the 1st)\n",
-    "3. **Effects persist over time** - past interventions continue to influence outcomes (behavioral habits change gradually)\n",
+    "2. **Effects persist over time** - past interventions continue to influence outcomes (behavioral habits change gradually, messages have carryover effects)\n",
+    "3. **Effects may saturate** (optional) - diminishing returns as exposure increases (10th message less impactful than the 1st)\n",
     "\n",
     "For a good introductory overview of transfer function models and intervention analysis, see {cite:p}`helfenstein1991use`.\n",
     "\n",
     "### Key Components\n",
     "\n",
-    "- **Saturation transforms**: Model diminishing returns using Hill, logistic, or Michaelis-Menten functions\n",
-    "- **Adstock (carryover) transforms**: Model persistence using geometric decay with configurable half-life\n",
-    "- **Baseline controls**: Include confounders and natural trends\n",
-    "- **{term}`Counterfactual` analysis**: Estimate effects by zeroing or scaling interventions\n",
+    "- **Transfer functions**: Transform the raw intervention variable to capture its dynamic relationship with the outcome. In the media mix modeling literature, two common transfer functions are:\n",
+    "  - **Adstock (carryover) transforms**: Model how effects persist over time using geometric decay with configurable half-life\n",
+    "  - **Saturation transforms** (optional): Model diminishing returns using Hill, logistic, or Michaelis-Menten functions when appropriate\n",
+    "- **Baseline controls**: Include confounders and natural trends in the regression\n",
+    "- **{term}`Counterfactual` analysis**: Estimate causal effects by zeroing or scaling interventions\n",
     "- **HAC standard errors**: Robust inference accounting for autocorrelation and heteroskedasticity\n",
     "\n",
+    "Transfer functions can be as simple as a distributed lag (adstock only) or can combine multiple transformations (e.g., saturation followed by adstock). The key is to match the functional form to the expected dynamics of the intervention.\n",
+    "\n",
     "### When to Use Graded Intervention Time Series\n",
     "\n",
     "Use this method when you have:\n",
     "- ✅ Time series data from a **single unit** (region, market, organization)\n",
     "- ✅ **Graded intervention** with varying intensity over time\n",
-    "- ✅ Reason to expect **saturation** (diminishing returns) or **carryover effects** (persistence)\n",
+    "- ✅ Reason to expect **carryover effects** (persistence over time), and optionally **saturation** (diminishing returns)\n",
     "- ✅ Baseline controls available for confounders\n",
     "\n",
     ":::{note}\n",