Forecasting with Two-way ANOVA in R - When interaction is absent.Rmd

---
title: "Forecasting with Two-Way ANOVA in R - When interaction is absent"
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---

## 1. Set up environment

```{r Load packages}
# Install pacman if needed
if (!require("pacman")) install.packages("pacman")

# load packages
pacman::p_load(pacman,
  tidyverse, openxlsx, ggpubr)
```

## 2. Load data

```{r Import sales dataset}
#Dataset is in datasets subfolder
(sales <- read.xlsx("datasets/Twowayanova.xlsx", sheet = "no_interaction"))
```

```{r Pivot data from wide to long dataset}
(sales_long <- sales %>% 
  pivot_longer(cols = starts_with("Price"), #columns to pivot to rows
               names_to = "price", #name the new column for the price variable
               values_to = "sales"))
```

## 3. Data Visualization

```{r Plot sales vs advertising colored by coupon}
#Plot using ggpubr
ggline(sales_long, x = "Advertising", y = "sales", 
       add = c("mean_se", "jitter"),
       color = "price", palette = "startrek",
       title = "Sales increase when ad spending increases at roughly the same rate",
       subtitle = "(no interaction as the curves in the graph are nearly parallel) ",
       legend.title = "Price Level"
       )
```

## 4. Two-way ANOVA (with replication)

```{r Two-way ANOVA table}
two_way_aov <- aov(sales ~ Advertising*price, data = sales_long)

#The anova table
summary(two_way_aov)
```

Since the p-values for the main effects advertising and then price are
small and the interaction (advertising*price) is very large.
Advertising and price factors (separately) impact sales. **Advertising
has an effect that is independent of price.**

## 5. AOV Model Diagnostics

```{r Plot the model diagnostics}
#Diagnostic plots
qqnorm(two_way_aov$residuals)
qqline(two_way_aov$residuals)
```

## 6. Forecast

```{r What we can expect in sales?}}
#What we can expect in sales when there is advertising vs. no advertising
(ads <- sales_long %>% 
  group_by(Advertising) %>% 
  summarize(sales_forecast = round(mean(sales),2),
            std_dev = round(sd(sales),2)))
#What we can expect in sales when the price is low medium or high
(price <- sales_long %>% 
  group_by(price) %>% 
  summarize(sales_forecast = round(mean(sales),2),
            std_dev = round(sd(sales),2)))
```

+----------------------+----------------------------+
| Predicted sales      | What we can expect in      |
| with:                | sales                      |
+:=====================+:===========================+
| High advertising     | 32.44                      |
+----------------------+----------------------------+
| Medium advertising   | 23.33                      |
+----------------------+----------------------------+
| Low advertising      | 19.44                      |
+----------------------+----------------------------+
| High price           | 16.33                      |
+----------------------+----------------------------+
| Medium price         | 24.78                      |
+----------------------+----------------------------+
| Low price            | 34                         |
+----------------------+----------------------------+

So in the case of sales, if we wanted to predict sales where both price
and advertising are significant and independent, we can use the
following equation:

> predicted sales = overall average + factor A effect (if significant) +
> factor B effect (if significant)

```{r Forecast when price is high and advertising is medium}
#Calculate the overall average
overall_avg <- round(mean(sales_long$sales),2)
paste("The overall sales average is", overall_avg, sep=" ")

#Calculate medium advertising effect
#Equivalent to (23.22 - 25.03704)
medium_adv_effect <- as.numeric(ads %>% filter(Advertising=="Medium") %>% select(sales_forecast)) - overall_avg

#Calculate high price effect
#Equivalent to (16.33 - 25.03704)
price_high_effect <- as.numeric(price %>% filter(price=="PriceHigh") %>% select(sales_forecast)) - overall_avg

#Calculate Forecast:
#predicted value = overall average + factor A effect (if significant) + factor B effect (if significant)
predicted_sales <- overall_avg + medium_adv_effect + price_high_effect

paste("Forecast when price is high and advertising is medium is:", predicted_sales, sep = " ")
```

## 7. Final Summary

The forecast equation we can use to predict with two-way ANOVA is:

> predicted sales = overall average + factor A effect (if significant) +
> factor B effect (if significant)

If a factor is **not significant** than the factor effect is **assumed
to be 0.**